Multiple Stochastic Integrals Construction of non-Gaussian Reflection Positive Generalized Random Fields Sergio Albeverio, Minoru W. Yoshida

no. 241

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungsgemeinschaft getragenen Sonderforschungsbereiches 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, August 2005 (rev. Fassung Januar 2006)

Multiple stochastic integrals construction of non-Gaussian reflection positive generalized random fields S. Albeverio1),2) , M. W. Yoshida3)∗ 17th Dec. 2005

1) Inst. Angewandte Mathematik, Universit¨ at Bonn, Wegelerstr. 6, D-53115 Bonn (Germany) 2) SFB611; BiBoS; CERFIM, Locarno; Acc. Architettura USI, Mendrisio; Ist. Mathematica, Universit` a di Trento 3) The Univ. Electro commun, Dept. Systems Engineering, 182-8585 Chofu-shi Tokio (Japan) Abstract Multiple stochastic integrals with respect to isonormal Gaussian process on Rd are used for the construction of reflection positive non-Gaussian S 0 (Rd ) valued random variables (random fields). This construction is then extended to a class of Hida distribution that includes the Wick powers of some classes of generalized Euclidean free fields as its elements also to random fields involving potential terms. Under general assumptions, it is proved that the reflection positivity (in the Osterwalder and Schrader sense) are satisfied by these Hida distributions and the random fields. The above newly developed general construction is used to discuss corresponding interesting examples. E.g., it provides a completely different proof of the reflection positivity of the Wick powers of generalized Euclidean free fields, as well as the construction of some new fields (without full Euclidean invariance but satisfying all other axioms, including space rotation invariance) and their Wick powers are the consequences of this probabilistic (and purely Euclidean) procedure.

MSC 2000: 60G15, 60H05, 60H40, 47B25, 81T08, 81T40. Keywords: Multiple stochastic integral,Wick powers, relativistic fields, Euclidean fields, Hida distribution.

0

Introduction

The main considerations performed in this paper are the following two: First, by making use of the multiple stochastic integrals by means of the isonormal Gaussian process on Rd (cf., e.g., Section 4.5 of [H]), we give a new general probabilistic framework to construct non-Gaussian S 0 (Rd ) valued random variables that possess the property of reflection positivity (in the sense of random variables). Where and throughout this paper we denote by d ∈ N the space-time dimension of the random fields and the ∗ e-mail

[email protected] fax +81 424 98 0541.

1

corresponding Euclidean fields, and by N, R, C the spaces of the natural numbers, the real numbers and the complex numbers respectively. This construction of non Gaussian S 0 (Rd ) valued random variables is extended to a class of Hida distributions (generalized random variables in a Kondratiev space), that includes the Wick powers of some class of generalized Euclidean free fields as its elements. We prove that the reflection positivity (in the weaker Osterwalder and Schrader sense (OS positivity), cf. (2.9)) is also satisfied by the elements of Hida distributions in this class (cf. Theorem 2.1). Also we define new reflection positive random fields involving potential terms (without full Eucliden invariance but satisfying all other axioms, cf. Remark 7, 8). Second we discuss interesting examples of interacting quantum fields. Examples provided here explain how the above mentioned general construction is applied. In particular, as an example of reflection positive Hida distributions, we prove by our probabilistic (and, thus, purely Euclidean methods) the reflection positivity of the Wick powers of generalized Euclidean free fields. We would like at this point to recall the fact that the Wick powers of generalized Euclidean free fields need not be S 0 (Rd ) valued random variable. For example, Nelson’s Euclidean free field φN is an S 0 (Rd ) valued random variable, which is a special case of the generalized Euclidean free fields discussed in the present work. The Wick power : φpN : can be defined as an S 0 (Rd ) valued random variable only for the cases where the space time dimensions d ≤ 2 and the powers p ∈ N or d = 3 and p = 1, 2, and for the other cases, e.g., for the higher dimensions d ≥ 4, : φ1N = φN is an S 0 (Rd ) valued random variable but : φpN : (p ≥ 2) are not S 0 (Rd ) valued random variables but actually Hida distributions (cf. Proposition 2, Remark 6), which define forms on the space (S) (cf. [H], [HKPS], [GrotS]). For this reason, the proof of the reflection positivity of the Wick powers of generalized Euclidean free fields can not be accomplished within the framework of the theory of S 0 (Rd )-random fields. Thus, precisely, the organization of this paper and the concepts underlying it will be the following. In section 1, we introduce an isonormal Gaussian process W ≡ {W (h), h ∈ L2 (Rd , λd )} on Rd which is defined on a complete probability space (Ω, F, P ), where λd denotes the Lebesgue measure on Rd . Subsequentially, by means of multiple stochastic integrals with respect to W we construct several S 0 (Rd ) valued random variables defined on the common probability space (Ω, F, P ). Starting from an abstract probability space has the advantage of building a framework for handling all relevant random variables, the probability laws of which correspond to the random fields and phenomena that we want to investigate (cf. [AY], [AFeY], [ABRY1], [ABRY2], [Y4]). This is the spirit of section 1.2 of [H]: ”In constructing or determining a probability space we first clarify the type of random phenomena to be analyzed and the probability structures to be investigated, and then fix Ω, F, P to fit in with these aims”. Following e.g., [AH-K], [AR], [Y1], [Y2], [Y3], we give a construction in this sprit which accomodate the isonormal Gaussian process on Rd (the Gaussian white noise) as well as other generalized white noises (cf. [AW], [AGW], [GrotS]). In this framework various quantities of physical importance can be expressed through suitably chosen random variables. By not fixing a Hilbert space ` a priori, but working with a wide probability space we have a framework which is flexible and permits to develop the analysis of very different phenomena. This probabilistic framework also fits in the spirit of the algebraic approach of quantum field theory where the various non-equilibrium local states are viewed in a common framework (cf. [Oj1] and the review [Oj2]): e.g., the representations of the algebra of the physical quantities characterized and defined through the Gel’fand-Naimark-Segal construction depend on the chosen states, and may be interpreted in the context of the probability theory through the concept of a family

2

of conditional probability measures defined on a given common (sufficiently widely chosen) probability space. Hence, precisely in section 1, for a given pseudo differential operator J on S(Rd ) ~ (τ, ξ) ~ ∈ R × Rd−1 , such that there exist m > 0, C > with a positive symbol j(τ, ξ), 0, α > 0 and ~ ≤ C(|τ |2 + |ξ| ~ 2 + m2 )− α2 , (0.1) 0 < j(τ, ξ) we define an S 0 (Rd ) valued random variable φJ on a complete probability space (Ω, F, P ) as the stochastic integral of the kernel J (using the same notation as for the corresponding pseudo differential operator J) with respect to W , that satisfies the following Z (Jϕ)(x)W (dx), ∀ϕ ∈ S(Rd ). S 0 < φ J , ϕ >S = Rd

~ by making use of multiple stochastic Next, assuming a bound ((1.18) below) for j(τ, ξ), integrals by means of W , we define the p-th Wick power : φpJ : (p ∈ N) of φJ as an S 0 (Rd ) valued random variable, and then specify the support properties of : φpJ : (p ∈ N) in Proposition 2. Such studies on the support properties of S 0 (Rd ) valued random variables are crucially important in the framework of the analysis on an abstract Wiener space (cf., e.g., [AY]), and it is the corresponding consideration for the random field defined by : φpJ : with the consideration on the sample path properties for the stochastic processes, in particular, for Brownian motion process (cf., e.g., section 2.2 of [H]), also in section 3.3 of [H] there is a discussion on the support properties of Gaussian random fields (the discussions given there cover the ~ 2 + m2 )− 12 ). results on φN which is equal to our : φpJ setting p = 1, j = (|τ |2 + |ξ| Corollary 3 will be used to prove the main results in section 2. n p In section 2, the moment functions (Schwinger functions) SJ,n ∈ S 0 ((Rd ) ), n ∈ N, of the S 0 (Rd ) valued random variable : φpJ : are defined under the assumption (1.18) for j: p SJ,n (ϕ1 ⊗ · · · ⊗ ϕn ) ≡ E[< ϕ1 , : φpJ :> · · · < ϕn , : φpJ :>]

ϕr ∈ S(Rd → C),

r = 1, . . . , n.

Then, in Definition 5, by making use of the structure of the above given Schwinger functions, for general J that need not satisfy the bound assumption (1.18) but satisfies n p 0 a regularity assumption (Assumption 1), we define SJ,n ∈ S6= ((Rd ) ), n ∈ N, which p formally corresponds to a ”moment function” of : φJ : (in this case : φpJ : need not be an S 0 (Rd ) valued random variable but only a Hida distribution, and the moments need not exist). Next, in Theorem 1, under the assumption that J satisfies (0.1) with Assumption 1, and that the corresponding S 0 (Rd ) valued random variable φJ satisfies the property p of reflection positivity (2.6), we show in a probabilistic manner that {SJ,n }n∈N defined in Definition 5 satisfies the Osterwalder-Schrader (O-S) positivity (2.9). Remark 7 is a consideration of new random fields satisfying reflection positivity but not satisfying Euclidean invariance. Remark 8 concerns the reflection positive random fields defined by adding a potential terms to the original random fields that satisfy the reflection positivity. Namely, in Remark 8, by assuming both the reflection positivity (2.6) and a new bound (2.20) for j, by making use of the S 0 (Rd ) valued random variable : φ2p J : (p ∈ N), it is shown that for each p ∈ N, Λ ≥ 0 and λ ≥ 0 the random field defined by the following probability measure µJ,p,Λ,λ ≡ µ on S 0 (Rd ) satisfies the property of reflection positivity (cf. [GRS], [AH-K], [AR], and for

3

another direction in this concern, [AY]): Z < ϕ1 , ω 0 > · · · < ϕn , ω 0 > µ(dω 0 ) S 0 (Rd )

≡

2p 1 E[e−λ < ϕ1 , φJ > · · · < ϕn , φJ >], ZΛ 2p

where ZΛ = E[e−λ ], amd IBΛ is the indicator function with BΛ ≡ {x ∈ Rd | |x| < Λ}. The considerations on such random fields in the contexts of Euclidean quantum field theory is new. In section 3, we examine some examples. An example having the correspondence to Remark 7 is Example I, where we study the random fields φJh , for Jh having the following form: 1 ∂2 (0.2) Jh = {− 2 + h(−∆d−1 + m2 )}− 2 , ∂t where h ∈ C ∞ (R+ → R+ ) is some given function such that h(x) ≥ C|x|β ,

∀x ∈ R+ ,

for some β > 0, C > 0, 2

(0.3) 2

∂ ∂ and ∆d−1 is the d − 1 dimensional Laplacian: ∆d−1 ≡ ∂x In 2 + · · · + ∂x 2. 1 d−1 Example I’, as a special case of Example I, we study the random fields φJh on Rd having Jh given by (0.2) which satisfy (0.3) with β ≥ d − 1. Remark 8 applies to such φpJh . In Example II, we discuss the case where

J = Jα ≡ (−

α ∂2 − ∆d−1 + m2 )− 2 , ∂t

α ∈ (0, 1].

In fact the generalized Euclidean free field, φJα and its Wick power : φpJα : characterized by Jα , are considered. By applying Theorem 2.1, the O-S positivity of the generalized Euclidean fields corresponding to them is proven and the corresponding Schwinger functions admit an analytic continuation. Before closing Introduction, for an adequate understanding of the concept of probabilistic proof of the reflection positivity of non-Gaussian random variables (and generalized random variables) indexed by S(Rd ), including Wick powers of the generalized Euclidean free fields as its special case, we should review the existing results and discussions having a connection with this subject: In the relativistic quantum field theory, generalized free fields were introduced by Greenberg [Gre] as natural generalizations of free fields satisfying all the axioms (resp. breaking part of them in a simple way), see, e.g. [Jo], [BogLogT], [Wi1,2] and [StWi] (p.105). Generalized free fields also occurs in a natural way in the discussion of certain ”structual questions” and ”triviality results”, see e.g. [FernFS], [DeA]. Wick powers of free fields have been studied extensively, see e.g. [StWi], [Bo], [LToll], [AFeY]. It is also natural to study in a similar way Wick powers of generalized free fields. This has been sketched in [Wi1,2]. A use of them for the construction of models of 2 space-time dimensional quantum fields has been done in [Ha1], [Ha2]. In [Klein], [Okabe], [SH] the O-S positivity of Euclidean generalized free fields is discussed. We have to point out that following the lines of [Wi1] and [Wi2], there is an alternative proof of the reflection positivity of the generalized Euclidean fields : φpJα : discussed in Example II: The ”Wightman functions” given by the analytic continuations of the {SJpα ,n }n∈N , can namely be identified with the known Wightman functions corresponding to the Wick powers of generalized free fields in relativistic quantum 4

field theory. In general starting from relativistic Hermitian scalar fields satisfying the Wightman axims and applying the fact that they imply the Osterwalder-Schrader axioms” (cf. [OS], [OS2], [Si], [Gl], [He], [OcaOtt]) one gets, in particular, the reflection positivity of the corresponding Euclidean fields. Thus, by the analytic continuation, once the Wick powers of generalized Euclidean free fields are identified with the Wick powers of generalized free fields in relativistic quantum field theory, then one can proceed using the fact that Wick powers of generalized free fields are Wightman fields (cf. [Wi3], [Br¨ u], [SS1,2] [Streater], [D¨ uR], [Licht1,2]), to prove the reflection positivity in Example II. However, as compared with this ”‘indirect proof”’, our probabilistic proof has the advantage of being direct (i.e. it is performed without analytic continuation from the Euclidean to the relativistic domain) and involves stochastic analytic estimates of interest by themselves. Our Theorem 1, Remarks 7 and 8 are in this way fully in accord with the Euclidean strategy in constructive quantum field theory, where one tries to verify the axioms for Euclidean field theory through arguments in the Euclidean regions. In this concern we may find a similarity between the present consideration and the ones which lead Zinoviev in [Z 1995] to modify the OS 1’ axiom (cf. (3.20)) to get a complete mathematical equivalence between Euclidean and relativistic axioms (see also [NaMu1,2,3]); ([OS2] does not have a complete equivalence in this sense as first pointed out in [Gl]).

1

Preliminaries

Let S(Rd ) be the Schwartz space of rapidly decreasing test functions equipped with the usual topology by which it is a Fr´echet nuclear space. Let S 0 (Rd ) be the topological dual space of S(Rd ). We denote the Fourier and Fourier inverse transform of a function ϕ ∈ S(Rd ) re√ R − −1x·ξ −1 −d 2 e ϕ(x)dx, F −1 [ϕ](ξ) = spectively by F[ϕ] and F [ϕ]: F[ϕ](ξ) = (2π) Rd √ R −1x·ξ d −d ϕ(x)dx for ϕ ∈ S(R ). We sometimes use the notation F[ϕ] = ϕ. ˆ (2π) 2 Rd e d By J we denote a pseudo differential operator on S(R ), for which we assume the following: There exist positive numbers m > 0, C > 0 and α > 0 such that α

~ ≤ C(|(τ, ξ)| ~ 2 + m2 )− 2 , 0 < j(τ, ξ)

~ ∈ R × Rd−1 , ∀(τ, ξ)

(1.1)

where ~ ≡ |(τ, ξ)|

q 2 , τ 2 + ξ12 + · · · ξd−1

τ ∈ R,

ξ~ ≡ (ξ1 , . . . , ξd−1 ) ∈ Rd−1 ,

and j ∈ S(Rd → R) is the symbol of the pseudo differential operator J: (Jϕ)(t, ~x) = F −1 (j ϕ)(t, ˆ ~x),

for

ϕ ∈ S(Rd ), (t, ~x) ∈ R × Rd−1 .

By the same notation J we denote the kernel corresponding to the pseudo differential operator as follows: Z (Jϕ)(t, ~x) = J(t − t0 , x1 − x01 , . . . , xd−1 − x0d−1 )ϕ(t0 , x01 , . . . , x0d−1 )dt0 dx01 · · · dx0d−1 , Rd

with the kernel defined by J(t, ~x) ≡ (F −1 j)(t, ~x). Throughout this paper we often use the notations ~ ∈ R × Rd−1 . ξ ≡ (τ, ξ)

x ≡ (t, ~x) ∈ R × Rd−1 , 5

(1.2)

α

We denote, in particular, the pseudo differential operator (−∆d +m2 )− 2 by J m,α , α > 0 (if there is no ambiguity, we often omit m > 0 and simply denote this by J α ), where ∆d denote the d-dimensional Laplace operator. By (1.2) we see that J m,α (x) ≥ 0 has the following regularlity (cf. [NaM3] and (3.13) in Example I): There exist some constants C1 , C2 > 0 (depending on m, α) such that ( C1 |x|−d+α for |x| < 1 m,α J (x) ≤ (1.3) C1 e−C2 |x| for |x| ≥ 1. Let η1 ∈ C0∞ (Rd ) be such that ( 0 ≤ η1 (x) ≤ 1

and

η1 (x) =

1

|x| ≤ 1

0

|x| ≥ 2.

(1.4)

and let ηk (x) = η1 ( xk ) ∈ C0∞ (Rd ), k = 1, 2, . . . . Also define ρ ∈ C0∞ (Rd ) as follows: 1 C exp(− ) |x| < 1 , (1.5) ρ(x) = 1 − |x|2 0 |x| ≥ 1 R where the constant C is taken to satisfy Rd ρ(x)dx = 1. Define ρk (x) = k d ρ(kx). (1.6) For the kernel of a pseudo differential operator J, of which symbol satisfies (1.1), we define a sequence of smooth functions Jk ∈ S(Rd ), k = 1, 2, . . . as follows: Z Jk (x) = J(y)ρk (x − y)dy = (ρk ∗ J)(x), x ∈ Rd . (1.7) Rd

Also, by the p ∈ N times multiplication of Jk we define, for yi ∈ Rd , i = 1, . . . , p, FkJ,p (x; y1 , . . . , yp ) = (ηk (x))p Jk (x − y1 ) · · · Jk (x − yp ).

(1.8)

Then the function FkJ,p is symmetric in the last p variables (y1 , . . . , yp ) and p+1

FkJ,p ∈ S((Rd )

FkJ,p (x; y1 , . . . , yp ) = 0

),

for |x| ≥ 2k.

(1.9)

For later convenience let us recall the following properties: ξ ρˆk (ξ) = ρˆ( ), k

|ˆ ρ(ξ)| ≤ 1,

ρˆ(0) = 1

(by (1.5)).

(1.10)

We also recall that ρˆk (ξ) converges to 1 uniformly on compact sets: For any M < ∞ and any > 0 there exists an N < ∞ such that 0 ≤ 1 − ρˆk (ξ) < , for any ξ such that |ξ| ≤ M and any k ≥ N . Now, suppose that on a complete probability space (Ω, F, P ) we are given an isonormal Gaussian process W = {W (h), h ∈ L2 (Rd ; λd )}, where λd denotes the Lebesgue measure on Rd (cf., e.g., [H], [Nu], [BaSeZh], [Y4]): W is a centered Gaussian family of random variables such that Z E[W (h)W (g)] = h(x) g(x)λd (dx), h, g ∈ L2 (Rd ; λd ). (1.11) Rd

We write

Z Wω (h) =

h(y)Wω (dy), Rd

6

ω∈Ω

with Wω (·) a Gaussian generalized random variable (in the notation of Hida calculus ˙ (y)dy cf. [HKPS]). for the Gaussian white noise Wω (dy) is often written as W For the pseudo differential operator J satisfying (1.1) and for each p ∈ N, we define a sequence of random variable :k φpJ,ω :, k ∈ N by means of the multiple stochastic integral (multiple Wiener integral) as follows (cf., e.g., [Nu], [AY]): Z p :k φJ,ω : (x) = FkJ,p (x; y1 , . . . , yp )Wω (dy1 ) · · · Wω (dyp ) x ∈ Rd . (1.12) (Rd )p

In the sequel, for p = 1 we use the simpler notation such that :k φ1J,ω : (x) = k φJ,ω (x). By this definition, from (1.9) we can take :k φpJ,ω : as a C0 (Rd → R) ≡ C0 (Rd )valued random variable. In fact, there exists a bounded open set Dk = {x | |x| < 2k} ⊂ Rd such that :k φpJ,ω : (x) = 0

for x ∈ Rd \ Dk

∀ω ∈ Ω,

and moreover, by the Kolmogorov’s continuity criterion the stochastic process {:k φpJ,ω : (x)}x∈Rd admits a continuous modification (cf., e.g.,[Nu]), which we also denote by :k φpJ,ω : (x): :k φpJ,ω : (·) ∈ C0 (Rd → R)

∀ω ∈ Ω.

(1.13)

For each a, b, d > 0, we define a linear subspace Bda,b of S 0 (Rd ) as follows: a

b

Bda,b = {(|x|2 + 1) 4 (−∆d + 1)+ 2 f : f ∈ L2 (Rd ; λd )},

(1.14)

where λ denotes the Lebesgue measure on R. Then Bda,b becomes a separable Hilbert space with the scalar product Z b b u, v ∈ Bda,b . < u|v >= J 1,a ((1 + |x|2 )− 4 u(x)) J 1,a ((1 + |x|2 )− 4 v(x))dx, Rd

(1.15) The following proposition 1 is easy to be provem in the same manner as in [Y4], where the case J = J α was considerd. Proposition 1 Let J be a pseudo differential operator satisfying (1.1), and :k φpJ,ω : be the random variable defined by (1.12). Then, for each k ∈ N and r ≥ 1 there exists Mk,r and the following holds: Z Z r p (1.16) :k φJ,ω : (x) dxP (dω) < Mk,r . Ω

Rd

Also for each k and l let U k,l (x1 , . . . , xl ) ≡ E (:k φpJ,· : (x1 )) · · · (:k φpJ,· : (xl )) , then U k,l ∈ C0 ((Rd )l → R).

(1.17)

Since (1.13) holds and C0 (Rd → R) ⊂ Bda,b (a, b, d > 0), we can consider the limit k → ∞ for :k φpJ,ω : (·) ∈ C0 (Rd → R) in Bda,b . As a consequence, we have the following Proposition 2, which can be proven in the same manner as in [Y4], where the case of J = J α was proved.

7

Proposition 2 integer p satisfy

Suppose that J satisfies (1.1) for some α > 0. Let the positive min{1,

2α 2a } + p × min{1, } > p, d d

(1.18)

for some a ∈ R, then the sequence of S 0 (Rd )-valued random variables {:k φpJ,ω :}k∈N has a limit in the space of Bda,b (b > d) valued random variables, hence the limit, denoted by : φpJ,ω :, is a Bda,b -valued random variable. Moreover the following hold: i) Z k :k φpJ,ω : − :m φpJ,ω : k2B a,b P (dω) = 0;

lim

k,m→∞

d

Ω

ii) There exists a P -null set N , a subsequence {:kj φpJ,ω :} of {:k φpJ,ω :} and a Bda,b -valued random variable : φpJ,ω : such that lim k :kj φpJ,ω : − : φpJ,ω : kB a,b = 0,

kj →∞

∀ω ∈ Ω \ N ;

d

For each ϕ ∈ S(Rd ) there exists a P -null set Nϕ and

iii)

S 0 ,S = lp,ω (ϕ)

∀ω ∈ Ω \ Nϕ ,

lim k S 0 ,S − S 0 ,S kL2 (Ω;P ) = 0,

k→∞

where Z lp,ω (ϕ)

= (Rd )p

Z (

ϕ(x) J(x − y1 ) · · · J(x − yp ) dx)

Rd

× Wω (dy1 ) · · · Wω (dyp );

(1.19)

iv) In particular for p = 1, (1.18) alwys holds for J satisfying (1.1), and a Bda,b valued random variable φJ,ω ≡: φ1J,ω : can be defined by ii). By Proposition 2-iv), since φJ,ω is an S 0 (precisely a Bda,b -) valued random variable, by Proposition 2-iii) the process ηk (x) < φJ,ω , Jk (x − ·) >S 0 ,S is a continuous process with parameter x ∈ Rd . On the other hand, by (1.13) since k φJ,ω (x) = R η (x)Jk (x − y)Wω (dy) is a continuous process with parameter x ∈ Rd , the two Rd k continuous processes (or C0 (Rd → R)-valued random variables) ηk (x) < φJ,ω , Jk (x − ·) > and k φJ,ω (x) are equivalent. We state this fact in Corollary 3-i) ( Corollary 3-ii) is a well known fact of the Wiener chaos decomposition). Corollary 3 Suppose that J satisfies (1.1). Let φJ ≡: φ1J : be the S 0 -valued random variable defined by Proposition 2-iv). Then the following statements are true: i) The continuous processes ηk (x) < Jk (x − ·), φJ,ω > and R 2 d k φJ,ω (x) = Rd ηk (x)Jk (x − y)Wω (dy) in L (Ω; P ) with parameter x ∈ R are equivalent in the following sense: there exists a P -null set N such that for all ω ∈ Ω \ N , k φJ,ω (x)

ii)

=< ηk (x)ρk (x − ·), φJ,ω >S,S 0

∀x ∈ Rd .

(1.20)

For any positive integer p p

:k φpJ,ω

[2] X

1 : (x) = p! m!(p − 2m)! m=0

m p−2m 1 − bk (x) , k φJ,ω (x) 2 8

(1.21)

where bk (x) is a function depending on J such that Z 2 2 bk (x) = ηk (x) Jk (x − z) dz. Rd

2

Reflection positive fields

We start with the consideration of the moment functions corresponding to the p-th Wick power of φJ in case where : φpJ : is a random variable. By Proposition 2, for J satisfying (1.18) for a power p under consideration, : φpJ : can be defined as an S 0 -valued random variable, and we can define the moment functions: for each p ∈ N (satisfying (1.18)) and n ∈ N, let spJ,n (ϕ1 ⊗ · · · ⊗ ϕn ) ≡ E[< ϕ1 , : φpJ :> · · · < ϕn , : φpJ :>], ϕr ∈ S(Rd → C),

(2.1)

r = 1, . . . n.

From (1.11) and (1.19), by setting 2 GJ = kernel of the pseudo differential operator with the symbol j(ξ) , then spn can be expressed by the explicit formula: p spJ,n (ϕ1 ⊗ · · · ⊗ ϕn ) =< SJ,n , ϕ1 ⊗ · · · ⊗ ϕn >S,S 0 ,

(2.2)

where p SJ,n (x1 , . . . , xn ) =

X

r(l)σn,p (GJ (xj − xi ) : l)

for np = even number

l∈Ln,p 0

otherwise,

(2.3) Ln,p is the totality of sequences l = (l1,2 , . . . , l1,n ; l2,3 , . . . , l2,n ; . . . ; ln−1,n ) such that li,j = 0, 1, . . . , p n X

l1,j = p,

l1,2 +

j=2

n X

l2,j = p,

l1,3 + l2,3 +

j=3

...,

k−1 X

li,k +

i=1

n X

j3,j = p, . . .

j=4 n X

lk,j = p, . . . ,

n−1 X

li,n = p.

(2.4)

i=1

j=k+1

and for each l ∈ Ln,p , σn,p (xj,i : l) = σn,p (x2,1 , . . . , xn,1 ; x3,2 . . . , xn,2 ; . . . ; xn,n−1 : l) is a monomial of xj,i ∈ R (i = 1, 2, . . . , n − 1, j = i + 1, . . . , n) defined by σn,p (xj,i : l) =

n n Y Y

(xj,i )li,j ,

i=1 j=i+1

with

also

r(l) =

Y

rij (l)

(2.5)

i 0 |GJ (x − x0 )| ≤ c(|~x − x~0 |l + 1) × (|t − t0 |r + |t − t0 |−r ), ∀x = (t, ~x) ∈ R × Rd−1 ,

x0 = (t0 , x~0 ) ∈ R × Rd−1 .

Note that Assumption 1 is satisfied by GJ α , (α > 0), by (1.3). Definition 5 Suppose that J satisfies (1.1) and Assumption 1. For each positive p integer p ∈ N we define SJ,n (n = 1, 2, . . . ) by (2.3). We call the continuous linear p dn functional SJ,n on S6= (R ) the n-th Schwinger function associated with : φpJ :. For p n = 0, we set SJ,0 ≡ 1. We note that under the regularity Assumption 1 for GJ , the fact that the above p 0 defined Schwinger functions SJ,n are elements of S6= (Rdn ) is obvious. Remark 6 (Hida distribution) In the framework of Hida calculus, Proposition 2 and Definition 5 are expressed as follows (here we use the notations of Hida calculus without any explanation, for these notations and notions cf. [GrotS] and [KLS]): d Suppose that J satisfies (1.1) for some α ∈ (0, d2 ), then for p ∈ N such that p < d−2α p (cf. (1.18)), the random variable defined in Proposition 2 is expressed by , where (only here) ω is the d-dimensional Gaussian white noise and Z (F J,p ϕ)(y1 , . . . , yp ) ≡ ϕ(x) J(x − y1 ) · · · J(x − yp ) dx, ϕ ∈ S Rd

10

(the ω in this context should of course not be confused with the generic point in Ω in d , are defined as Hida the rest of the paper). Also, for p ≥ d−2α distributions in (S)−1 , which are the continuous functionals on (S)1 such that >(S)−1 ,(S)1 Z = p! (F J,p ϕ)(y1 , . . . , yp )ψ(y1 , . . . , yp )dy1 · · · dyp , Rdp

for ψ ∈ S((Rd )p ), where I(ψ) =. Moreover (cf. [KLS], [GrotS]), it is possible to define as an element of (S)−1 for each x ∈ Rd , where δ{x} is the Dirac point measure at x, and the following holds Z ϕ(x)dx =, ϕ ∈ S. Rd

Namely, the notion : φpJ : (and the Shwinger functions corresponding to it) given in Definition 5 has the expression by means of a Hida distribution: . Definition 5 says that generally speaking, an element of (S)−1 , , only defines a form on (S), however, for J satisfying (1.1) and Assumption 1, the expectation E( · · · ) 0 has a meaning and defines an element of S6= (Rdn ).

To state the reflection positivity of fields on S(Rd ) we need the notion of the time reflection operator. The time reflection operator θ is defined by θ(t, ~x) = (−t, ~x), as a map from Rd into Rd . We also denote by θ its induced action on elements n fn ∈ S((Rd+ ) ) such that θfn (x1 , · · · , xn ) = fn (θ−1 x1 , · · · , θ−1 xn ). Assumption 2 Assume that J satisfies (1.1) for some α > 0. Suppose also that a,b the Bd -valued random variable φJ,ω on (Ω, F, P ) defined by Proposition 2-iv) satisfies the property of reflection positivity in the sense of random variables (cf., e.g., Definition 5 of [AGW]): For anr ∈ C, ϕn,r ∈ S(Rd+ → C), r = 1, . . . , Nn , l = 1, . . . , n, n = 1, . . . , N l (Nn , N ∈ N) and c ∈ C, n X Nn N X o ¯n,r a¯nr < θϕ¯n,r ¯ E n , φJ > · · · < θϕ1 , φJ > + c n=1 r=1

×

Nn N X nX

o n,r anr < ϕn,r , φ > · · · < ϕ , φ > + c ≥ 0. J J n 1

(2.6)

n=1 r=1

Examples of J by which Assumption 2 is satisfied are given in the next section (cf. Example I, Remark 9 and Example II). We remark that for φJ,ω satisfying Assumption 2, the corresponding Schwinger functions {s1J,n }n∈N∪{0} satisy the following O-S positivity not only for fn ∈ S+ (Rdn ), but also for fn ∈ S((Rd+ )n ): N X N X

n

1 Sn+m (θfn∗ ⊗ fm ) ≥ 0 ∀f0 ∈ C, fn ∈ S((Rd+ ) ),

n=0 m=0

11

n = 1, . . . , N,

(2.7)

where fn∗ (x1 , . . . , xn ) = fn (xn , . . . , x1 ). Now, we restrict the space of test functions to S6= (Rdn ), and extend the results on reflection positivity to the case of Wick powers. Theorem 2.1 Assume that for a given J, Assumption 1 and Assumption 2 are p satisfied. Let {SJ,n }n∈N be the sequence of Schwinger functions defined by Definition 5. Then p 0 SJ,n ∈ S6= (Rdn ) (2.8) p and {SJ,n } satisfies the O-S positivity: N X N X

p SJ,n+m (θfn∗ ⊗ fm ) ≥ 0

for all

f0 ∈ C, fn ∈ S+ (Rdn ), n = 1, . . . , N.

n=0 m=0

(2.9) In particular, if α > 0 in (1.1) satisfies α ≥ d2 , then for every p ∈ N, : φpJ : is an S 0 (Rd )-valued random variable and satisfies the reflection positivity in the sense of random variables, namely it satisfies (2.6) with the replacement of φJ by : φpJ . Proof. (2.8) is a direct consequence of the regularity Assumption 1 for GJ . In order to prove (2.9), it suffices to show that N N X X

p (θfn∗ ⊗ fm ) ≥ 0 SJ,n+m

(2.10)

n=0 m=0

for fn of the special form fn = such that

PNn

r=1

n,r dn anr ϕn,r 1 ⊗ · · · ⊗ ϕn ∈ S+ (R ), n = 0, . . . , N ,

ϕn,r ∈ C0∞ (Rd → R) ∩ S(Rd+ → R), l

l = 1, . . . , n,

r = 1, . . . , Nn ,

(2.11)

(the general case follows by a continuity and density argument). We prove (2.10) only for N = 2, N1 = 0 and N2 = 1 (for other N and Nn the proof can be carried out similarly, and it will be briefly explained after finishing the proof for these general cases). For each k ∈ N define GJ,k (xj , xi ) by Z GJ,k (xj , xi ) = FkJ,2 (y; xj , xi )dy, (2.12) Rd

where FkJ,2 is defined by (1.8). Using GJ,k (xj , xi ) for given p we define p SJ,n,(k) (x1 , . . . , xn ) with the replacement of GJ (xj −xi ) by GJ,k (xj , xi ) in (2.3). Then for f = ϕ1 ⊗ ϕ2 ∈ S+ (R2d → R) such that ϕi ∈ C0∞ (Rd → R) ∩ S(Rd+ → R), i = 1, 2, by Fubini’s lemma and then Lebesgue’s convergence theorem we see that, for ∀c ∈ R, p p p SJ,4 (θf ⊗ f ) + cSJ,2 (θf ) + cSJ,2 (f ) + c2 n o p p p = lim SJ,4,(k) (θf ⊗ f ) + cSJ,2,(k) (θf ) + cSJ,2,(k) (f ) + c2 . k→∞

12

(2.13)

On the other hand by (1.11), (1.12), (1.16) and Fubini’s lemma p p p SJ,4,(k) (θf ⊗ f ) + cSJ,2,(k) (θf ) + cSJ,2,(k) (f ) + c2 = E {< θϕ2 , :k φpJ :>< θϕ1 , :k φpJ :> +c}{< ϕ1 , :k φpJ :>< ϕ2 , :k φpJ :> +c} n Z Z o =E ϕ2 (θx) :k φpJ : (x)dx ϕ1 (θx) :k φpJ : (x)dx + c Rd Rd Z n Z o × ϕ1 (x) :k φpJ : (x)dx ϕ2 (x) :k φpJ : (x)dx + c d Rd Z R = ϕ1 (x)ϕ1 (x0 )ϕ2 (y)ϕ2 (y0 ) R4d h i ×E :k φpJ : (x) :k φpJ : (θx0 ) :k φpJ : (y) :k φpJ : (θy0 ) dxdx0 dydy0 Z h i +c ϕ1 (x)ϕ2 (y)E :k φpJ : (x) :k φpJ : (y) dxdy 2d ZR h i +c ϕ1 (x)ϕ2 (y)E :k φpJ : (θx) :k φpJ : (θy) dxdy + c2 . (2.14) R2d

By (2.13) and (2.14), in order to show that the LHS of (2.13) is not less than 0 it suffices to prove that the RHS of (2.14) is not less than 0 when k is large enough. For above ϕ1 and ϕ2 let δ be such that δ ≡ min d(supp[ϕ1 ], ∂Rd+ ), d(supp[ϕ1 ], supp[ϕ2 ]) where d(A, B) represents the (Euclidean) distance between two sets A and B, and M ∂Rd+ is the boundary of Rd+ . For each M ∈ N we take {AM j }j=1,...,M and {Bj }j=1,...,M as coverings of supp[ϕ1 ] respectively supp[ϕ2 ], such that d d(AM j , ∂R+ ) > δ −

1 , M

SM

j=1

M [

M d(AM j , Bj ) > δ −

AM j and

SM

AM j ⊃ supp[ϕ1 ],

j=1

1 , M

∀j, ∀j 0 = 1, . . . , M ;

BjM are compact,

M 0 AM j ∩ Aj 0 = ∅ (j 6= j ),

j=1

|supp[ϕ1 ]| − 1 /M ≤ |AM j | ≤ |supp[ϕ1 ]| + 1 /M, M [

BjM ⊃ supp[ϕ2 ],

∀j = 1, . . . , M ;

BjM ∩ BjM0 = ∅ (j 6= j 0 ),

j=1

|supp[ϕ2 ]| − 1 /M ≤ |BjM | ≤ |supp[ϕ2 ]| + 1 /M, ∞ [ M \

∞ [ M \

AM j = supp[ϕ1 ],

M =1 j=1

∀j = 1, . . . , M ;

BjM = supp[ϕ2 ],

M =1 j=1

where |A| denots the volume (evaluated by Lebesgue measure) of a set A. M M For each M and j we denote some points in the partition AM j and Bj by xj 4

and yjM respectively. By (1.17) we know that U k,4 ∈ C0 ((Rd ) → R) and U k,2 ∈ 2

C0 ((Rd ) → R), and the RHS of (2.14) is a pointwise limit of Riemann sums, i.e.: The RHS of (2.14) = lim Ek (M ) + c2 , M →∞

13

(2.15)

where Ek (M )

=

M X M X M X M X i=1

× U + c

k,4

i0 =1

j=1

M M ϕ2 (yjM0 )ϕ1 (xM i0 )ϕ1 (xi )ϕ2 (yj )

j 0 =1

M M M M M M (θyjM0 , θxM i0 , xi , yj )|Ai0 ||Bj 0 ||Ai ||Bj |

M X M X

M k,2 M M ϕ1 (xM (xi , yjM )|AM i )ϕ2 (yj )U i ||Bj |

i=1 j=1

+ c

M X M X

M k,2 M M ϕ1 (xM (θyjM0 , θxM i0 )ϕ2 (yj 0 )U i0 )|Ai0 ||Bj 0 |.

i=10 j 0 =1 1 < δ and supp[ϕ1 ] ∪ supp[ϕ2 ] ⊂ It can be seen that if k and M satisfy k1 + M {x | |x| < k}, then Ek (M ) + c2 ≥ 0. (2.16)

For this purpose, by Proposition 1 and again by Fubini’s lemma we write Ek (M ) + c2 as follows: Ek (M ) + c2 n X M X M o p M M M =E ϕ2 (yjM0 )(:k φpJ : (θyiM0 ))ϕ1 (xM )(: φ : (θx ))|A ||B | + c 0 0 0 0 k J i i i j j 0 =1 i0 =1 M X M nX

×

o p p M M M M M ϕ1 (xM )(: φ : (x ))ϕ (y )(: φ : (y ))|A ||B | + c . (2.17) k J 2 j k J i i j i j

i=1 j=1

To show that (2.16) holds we crucially use (1.20) and (1.21). We give the detailed proof of (2.16) for p = 2 (for p ≥ 3 the proof of (2.16) is similar). Since k is taken to satisfy supp[ϕ1 ] ∪ supp[ϕ2 ] ⊂ {x | |x| < k}, by (1.4) and the definition of ηk we have M M M ηk (xM i ) = ηk (θxi ) = ηk (yj ) = ηk (θyj ) = 1,

i, j = 1, . . . M.

By this for i, j = 1, . . . , M , bk (xM i )

=

bk (θxM i )

=

bk (yjM )

=

bk (θyjM )

Z = bk ≡

2 Jk (· − z) dz.

Rd

Hence, for p = 2 by (1.20) and (1.21) we have M X M X

2 M M 2 M M M ϕ1 (xM i )(:k φJ : (xi ))ϕ2 (yj )(:k φJ : (yj ))|Ai ||Bj | + c

i=1 j=1

=

M X M X

n o M M M M 2 ϕ1 (xM − bk (xM k φJ (xi ) i )ϕ2 (yj )|Ai | |Bj | i )

i=1 j=1

n o 2 × k φJ (yjM ) − bk (yjM ) + c =

M X M X

aij < fi , φJ >< fi , φJ >< gj , φJ >< gj , φJ >

i=1 j=1

+

M X

ai < fi , φJ >< fi , φJ > +

i=1

M X j=1

14

a0j < gj , φJ >< gj , φJ > +a0 , (2.18)

where M M M aij = ϕ1 (xM i )ϕ2 (yj )|Ai | |Bj |,

fi (·) = ρk (xM i − ·),

M X M ai = −bk ϕ1 (xM ϕ2 (yjM )|BjM |}, i )|Ai |{

gj (·) = ρk (yjM − ·),

M X M a0j = −bk ϕ2 (yjM )|BjM |{ ϕ1 (xM i )|Ai |},

j=1

i=1 2

a0 = c + (bk )

M X M X

M M M ϕ1 (xM i )ϕ2 (yj )|Ai | |Bj |.

i=1 j=1

Also since ρ(θx − y) = ρ(x − θy) we have M X M X

2 M M M ϕ2 (yjM )(:k φ2J : (θyjM ))ϕ1 (xM i )(:k φJ : (θxi ))|Ai ||Bj | + c

j=1 i=1

= a0 +

M X M X

aij < θgj , φJ >< θgj , φJ >< θfi , φJ >< θfi , φJ >

i=1 j=1

+

M X

a0j < θgj , φJ >< θgj , φJ > +

j=1

M X

ai < θfi , φJ >< θfi , φJ > . (2.19)

i=1

For ϕ1 ⊗ϕ2 ∈ S+ (R2d → R) such that ϕi ∈ C0∞ (Rd → R)∩S(Rd+ → R), i = 1, 2, d 1 1 M since ρk (xM i −·), ρk (yj −·) ∈ S(R+ → R) for every i, j = 1, . . . , M when k + M < δ, supp[ϕ1 ] ∪ supp[ϕ2 ] ⊂ {x | |x| < k} (cf. (1.6)), by (2.18) and (2.19) from (2.6) the RHS of (2.17) is non-negative for such k and M , i.e. (2.16) holds (note that φJ,ω satisfies the reflection positivity in the sense (2.6) of random variables); by (2.16) and (2.15) the RHS of (2.14) is non-negative for such k; by this the LHS of (2.13) is nonnegative and hence (2.10) has been proved for N = 2, N1 = 0, N2 = 1 and p = 2. The other cases can be proved in a similar way: RHS of the formula (2.18) corresponding to (cf. (2.14)) < ϕ1 , :k φpJ :> · · · < ϕn , :k φpJ :> +c becomes a polymomial of < f, φJ >, f ∈ C0∞ (Rd → R) ∩ S(Rd+ → R), with the power np, and the non-negativity of (2.17) corresponding to the general cases are proved through the same arguments performed above. Remark 7 (Modification of the case α ≥ d2 ) Suppose that J satisfies the following: There exist m > 0, C > 0, β > 0 and γ > 12 such that β≥

d−1 , 2γ − 1

and n o− γ2 ~ ≤ C |τ |2 + (|ξ| ~ 2 + m2 )β 0 ≤ j(τ, ξ) ,

~ ∈ R × Rd−1 . ∀(τ, ξ)

(2.20)

By the same argument as for Proposition 2, under the condition (2.20) we easily see that for every p ∈ N, : φpJ : can be defined as an S 0 (Rd ) valued random variable taking values in the same region (sub space of S 0 (Rd )) in which : φpJ α : with α ≥ d2 takes 15

values. As a consequence, for : φpJ α : (p ∈ N) with J satisfying (2.20), Proposition 2 holds for every p ∈ N. Also, by (1.20) and (1.21), for ϕ ∈ S(Rd+ → R) the random variable < ϕ, : T p φJ,ω :>S,S 0 ∈ q≥1 Lq (Ω; P ) is σ{< ϕ, φJ,ω >S,S 0 |ϕ ∈ S(Rd+ → R)} measurable, where σ{< ϕ, φJ,ω >S,S 0 |ϕ ∈ S(Rd+ → R)} is the sub σ-field of F generated by the random variables < ϕ, φJ,ω >S,S 0 , ϕ ∈ S(Rd+ → R). Since, if we assume in addition that φJ satisfies (2.6), the reflection positivity in the sense of random variables, then by the Lebesgue’s convergence theorem (on the probability space (Ω, F, P )) every σ{< ϕ, φJ,ω >S,S 0 |ϕ ∈ S(Rd+ → R)} measurable L2 (Ω; P ) random variable satisfies the reflection positivity in the sense of random variables, we see that for every p ∈ N and n ∈ N the random variable < ϕ1 , : φpJ,ω :> · · · < ϕn , : φpJ,ω :>, ϕi ∈ S(Rd+ → R), i = 1, . . . , n, satisfies the reflection positivity , namely it satisfies (2.6) by replacing φJ by : φpJ :. In such a way, if : φpJ : is an S 0 -valued random variable, then the proof of reflection positivity becomes simpler, but for : φpJ : that is not a random variable but only a Hida distribution, then in order to get the result of O-S positivity (the O-S axioms [OS1,2] expression of reflection positivity) we should add a regularity assumption for GJ (cf. Assumption 1, cf. also [NaMu1,2,3] for more singular ”Shwinger functions” corresponding to hyperfunctions ). Remark 8 (Adding potential terms) For A ∈ B(Rd ), let χA denote the ind dicator function, where B(R ) is the Borel σ-field of Rd . Assume that J satisfies (2.20) or (1.1) with α ≥ d2 . Then Proposition 2-iii) holds not only for ϕ ∈ S but also for χA , precisely, for every p ∈ N it is possible to show that Z lim k :k φpJ : (x)dx − lp,ω (χA )kL2 (Ω;P ) = 0 ∀A ∈ B(Rd ), (2.21) k→∞

A

where lp,ω (χA ) is the random variable defined by (1.19) by setting ϕ = χA (χA is not an element of S but (1.19) is well defined by (1.11)). Because of (2.21), we will denote lp,ω (χA ) by . Now, for L > 0, let ΛoL,+ ≡ {(t, ~x) ∈ R × Rd−1 : |(t, ~x)| < L,

t > 0},

ΛoL,− ≡ {(t, ~x) ∈ R × Rd−1 : |(t, ~x)| < L,

t < 0},

ΛL,+ ≡ {(t, ~x) ∈ R × R

d−1

: |(t, ~x)| < L,

t ≥ 0},

ΛL,− ≡ {(t, ~x) ∈ R × Rd−1 : |(t, ~x)| < L,

t ≤ 0},

ΛL ≡ {(t, ~x) ∈ R × R

d−1

: |(t, ~x)| < L},

Then, through a direct calculation by using (1.11) and (1.19), for any p ∈ N we have k − kL2 (Ω;P ) = 0, k − kL2 (Ω;P ) = 0, Hence, for P − a.e. ω we see that =, = . (2.22) Next, recall that for the Gaussian random fields defined by stochastic integrals and multiple stochastic integrals of Jϕ (ϕ ∈ S), such that J satisfies (1.1) with α ≥ d2 , 16

the following evaluation holds (cf. Th. 1.22 of [Si], for applications see, e.g., [AY], [AFY]): m (q ≥ 2), kΦkLq (Ω;P ) ≤ (q − 1) 2 kΦkL2 (Ω;P ) , where, for fl ∈ S(Rd → R), l = 1, . . . , m, m ∈ N, Z Φ= (Jf1 )(x1 ) · · · (Jfm )(xm )W (dx1 ) · · · W (dxm ). (Rd )m

The above is also true for J satisfying (2.20). By making use of this bound, through the same arguments as in Theorem V.7 in [Si], for J satisfying (2.20) or (1.1) with α ≥ d2 , we have for any λ ≥ 0 and p ∈ N, o exp{−λ },

o exp{−λ },

exp{−λ } ∈

\

Lq (Ω; P ).

(2.23)

q≥1

By (2.22) and (2.23) we have \ 2p o o Lq (Ω; P ) 3 exp{−λ } × exp{−λ } q≥1

=

exp{−λ },

P − a.s..

(2.24)

d o Since, exp{−λ } is σ{< ϕ, φJ >S,S 0 : ϕ ∈ S(R+ → R)} measurable, and 2p o o exp{−λ } = exp{−λ }

(2.25)

is σ{< ϕ, φJ >S,S 0 : ϕ ∈ S(Rd− → R)} measurable, where Rd− ≡ {x ∈ Rd | x = (t, ~x) ∈ R×Rd−1 , t < 0}, if we assume in addition that φJ satisfies Assumption 2 (the reflection positivity in the sense of random variables), then by the result of Remark 7 ( the fact that σ{< ϕ, φJ,ω >S,S 0 |ϕ ∈ S(Rd+ → R)} measurable L2 (Ω, P ) random variable is reflection positive) , from (2.24) and (2.25) we see that the following holds: For anr ∈ C, ϕn,r ∈ S(Rd+ → C), r = 1, . . . , Nn , l = 1, . . . , n, n = 1, . . . , N l (Nn , N ∈ N) and c ∈ C, Nn N X nX o 2p ¯n,r E e−λ a¯nr < θϕ¯n,r ¯ n , φJ > · · · < θϕ1 , φJ > + c n=1 r=1

×

Nn N X nX

anr

···

+ c /ZΛ ≥ 0,

n=1 r=1

where

2p

ZΛ = E[e−λ ].

3

Examples

In this section, we apply Theorem 1 and Remark 8 to three examples.

17

(2.26)

Example I. For given (space time) dimension d ∈ N and mass m > 0, suppose that the symbol of the pseudo differential operator J is given by 1 ~ ∈ R × Rd−1 , ~ ≡ τ 2 + h(|ξ| ~ 2 + m2 ) − 2 , (τ, ξ) (3.1) jh (τ, ξ) where h ∈ C ∞ (R → R+ ) is a given function that satisfies h(x) ≥ C|x|β ,

∀x ∈ R,

(3.2)

for some β > 0, C > 0. In this section, these pseudo differential operators with symbols having the form given by (3.1) will be denoted by Jh . Then Jh is expressed by Jh = −

− 1 ∂2 + h(−∆d−1 + m2 ) 2 , 2 ∂t

where ∆d−1 is the d − 1 dimensional Laplacian:

∆d−1 =

(3.3) ∂2 ∂x21

+ ··· +

∂2 . ∂x2d−1

By

(3.1) and (3.2), since jh satisfies (1.1), from Proposition 2-iv), φJh is defined as an S 0 (Rd )-valued random variable. In order to apply Theorem 1 to φJh and its Wick powers, we have to show that 1◦ ) φJh satisfies the reflection positivity in the sense of random variable; 2◦ ) the integral kernel GJh satisfies the regularity Assumption 1. By modifying a standard discussion for Euclidean invariant random fields introduced by [N1,2] (cf. also [Si]), we can show 1◦ ) in the following way Let Hh ≡ completion of S(Rd → R) with respect to the norm Z 12 ~ ϕ(τ, ~ 2 dτ dξ~ , ϕ ∈ S(Rd → R). kϕkHh ≡ (jh (τ, ξ) ˆ ξ)) R×Rd−1

Then Hh ⊂ S 0 (Rd → R) is a Hilbert space equipped with the inner product given by (where we use the complex Hilbert space notation) Z ~ 2 ϕ(τ, ~ ψ(τ, ˆ ξ)dτ ~ dξ. ~ (ϕ, ψ)Hh = (jh (τ, ξ)) ˆ ξ) (3.4) Rd

Similar to the discussion for (2.21), < φJh , ϕ > can be defined as an L2 (Ω; P ) random variable for any ϕ ∈ Hh . Let Mh ≡ ϕ ∈ Hh | supp[ϕ] ⊂ {(t, ~x) ∈ Rd | t ≤ 0} , Nh ≡ ϕ ∈ Hh | supp[ϕ] ⊂ {(t, ~x) ∈ Rd | t = 0} . Now, take ϕ ∈ Hh such that supp[ϕ] ⊂ {(t, ~x) ∈ Rd | t > 0},

(3.5)

and let ϕ˜ be the orthogonal projection of ϕ onto Mh . Then (cf. Theorem 5 of [N2]) < ψ, (−

∂2 ∂2 2 −1 +h(−∆ +m )) ϕ ˜ >=< ψ, (− +h(−∆d−1 +m2 ))−1 ϕ >, ∀ψ ∈ Mh . d−1 ∂t2 ∂t2

In the above equation, by setting ψ ∈ C0∞ (Rd− ) ⊂ Mh with supp[ψ] ⊂ {(t, ~x)| t < 0}, we see that (−

∂2 ∂2 2 −1 + h(−∆ + m )) ϕ ˜ = (− + h(−∆d−1 + m2 ))−1 ϕ d−1 ∂t2 ∂t2

18

on {(t, ~x)| t < 0}. (3.6)

2

∂ 2 But the operator − ∂t 2 +h(−∆d−1 +m ) keeps the support unchanged on the time axis ∂2 2 −1 of the functions. Thus, by (3.5) and (3.6) (by operating (− ∂t 2 + h(−∆d−1 + m )) both side of (3.6)) we see that

supp[ϕ] ˜ ⊂ {(t, ~x) | t = 0} i.e.

ϕ˜ ∈ Nh 0

for

ϕ ∈ Hh

satisfying (3.5).

By this it follows that the S (R → R)-valued random variable φJh is an S 0 (Rd−1 → R)-valued Markov process with the parameter t ∈ R, namely for ϕ ∈ Hh satisfying (3.5) the following holds: d

E[< φJh , ϕ > | σ(Mh )] = E[< φJh , ϕ > | σ(Nh )],

P − a.s.,

(3.7)

where σ(Mh ) and σ(Nh ) are the σ-fields generated by the random variables {< φJh , ψ > | ψ ∈ Mh } and {< φJh , ψ > | ψ ∈ Nh } respectively. Next, following the proof of Theorem 1 in [N1] (cf., also, the proof of Proposition III.4-(a) of [Si], for more general situations cf. Theorem 2.14 of [Mizohata]) we use the fact that ψ ∈ Nh admits a representation such that ψ(t, ~x) =

n X

gm (~x)δt=0 ,

for suitable gm ∈ S 0 (Rd−1 → R), m = 1, . . . , n,

n ∈ N,

m=1

(3.8) where δt=0 is the Dirac point measure at t = 0. From (3.8), by using (1.11), (1.19) and (3.1), for ϕ ∈ Hh satisfying (3.5), passing through a direct calculation we see that E[< φJh , ϕ >< φJh , ψ >] = E[< φJh , θϕ >< φJh , θψ >],

∀ψ ∈ Nh ,

(3.9)

where θ is the time reflection operator defined in the previous section. Hence, for ϕ ∈ Hh satisfying (3.5) we see that the following holds (”reflection property” p.101 in [N1]): E[< φJh , ϕ > | σ(Nh )] = E[< φJh , θϕ > | σ(Nh )],

P − a.s..

2

(3.10)

∂ 2 −1 need not satisfy the We should remark that the operator (− ∂t 2 + h(−∆d−1 + m )) maximum principle, which is used in the proof of Theorem 1 of [N1] where the case of ∂2 2 −1 (− ∂t (the Nelson’s Euclidean free field) is discussed. For random 2 − ∆d−1 + m ) fields that are not necessarily Euclidean invariant to prove the ”reflection property” we only need the fact that the expression (3.8) holds for ϕ ∈ Nh (i.e. we only need the regularity on the time axis, cf. Remark 9, below). By (3.7) and (3.10), we have 1◦ ) immediately: In fact, for ϕ ∈ Hh satisfying (3.5) we get

E[< φJh , ϕ >< φJh , θϕ >] h i = E E[< φJh , ϕ > | σ(Mh )] < φJh , θϕ > h i = E E[< φJh , ϕ > | σ(Nh )] < φJh , θϕ > h i = E E E[< φJh , ϕ > | σ(Nh )] < φJh , θϕ > σ(Nh ) h i = E E[< φJh , ϕ > | σ(Nh )]E[< φJh , θϕ > | σ(Nh )] ≥ 0. 19

Since, the above calculation is valid for the replacement of < φJh , ϕ >< φJh , θϕ > by the random variable inside of the expectation of the LHS of (2.6), we have thus proven 1◦ ). To show that 2◦ ) holds, we remark that ~ (t, ~x) GJh (t, ~x) ≡ F −1 j 2 (τ, ξ) 1 Z √ −1 ~ 2 + m2 ) −1 dτ (~x) √ e −1t·τ τ 2 + h(|ξ| = Fd−1 2π R r √ π 1 2 ~2 −1 q e− h(|ξ| +m )|t| (~x), (3.11) = Fd−1 2 ~ 2 + m2 ) h(|ξ| −1 where Fd−1 denotes the Fourier inverse transform with respect to the variable ξ~ ∈ d−1 R . For β > 0 given in (3.2), define

r = r(d, β) ≡ min{r0 ∈ N | βr0 ≥ d}.

(3.12)

Since, √ 2 ~2 e− h(|ξ| +m )|t|

1 q

∞ n 1 X i−1 1 ~ 2 + m2 ) 2 |t|n ~ 2 + m2 ) 2 h(|ξ| h(|ξ| n! n=0

=

h

≤

|t|−r

~ 2 + m2 ) h(|ξ|

r+1 1 ~ 2 + m2 ) − 2 , h(|ξ| r!

by (3.11) we see that |GJh (t, ~x)| ≤ C|t|−r , where C=

1 r!

Z

∀~x ∈ Rd−1 ,

r+1 ~ 2 + m2 ) − 2 dξ~ < ∞. h(|ξ|

(3.13)

(3.14)

Rd−1

Hence, GJh satisfies Assumption 1 for r > 0, C > 0 given by (3.12) and (3.14) resp., and l = 0. This then proves 2◦ ). As a consequence, by applying Theorem 1 we see that (2.9) holds for φJh and its Wick powers, that are not necessarily random variables. Remark 9 (Reflection property) that jh has the form

As a generalization of Example I, suppose

k ~ ≡ τ 2 + h(|ξ| ~ 2 + m2 ) − 2 , jh (τ, ξ)

for some k ∈ N,

where h ∈ C ∞ (R → R+ ) satisfies (3.2) as before. Then, the discussion for the derivation of (3.7) is also valid for this general case (i.e., for any k ∈ N): for ϕ ∈ Hh satisfying (3.5) we have E[< φJh , ϕ > | σ(Mh )] = E[< φJh , ϕ > | σ(Nh )],

P − a.s..

(3.70 )

However, for k ≥ 2 we do not have the expression (3.8) for ψ ∈ Nh any more, namely 0 ψ ∈ Nh may have a component such that g(~x)δt=0 , where g(~x) ∈ S 0 (Rd−1 → R) and 0 δt=0 is the derivative of the Dirac point measure δt=0 , hence (3.9) need not hold and we do not have the ”reflection property” (3.10) (for these discussions cf. [N2]). 20

As a consequence, even though the Markov property (3.7’) holds, we can not get the reflection positivity for φJh with Jh given above (k ≥ 2). In terms of probability theory, this discussion can be expressed as follows: For the case where k ≥ 2, the S 0 (Rd )-valued random variable φJh satisfies a generalized Markovian property property with respect to σ(Nh ) and it is also an S 0 (Rd−1 )-valued stochastic process with the parameter t ∈ R, but it is not an S 0 (Rd−1 )-valued (symmetric) Markov process, because the σ-field generated by the random variables with test functions ψ having the form (3.8) is exactly a sub σ-field of σ(Nh ). Namely the information available through S 0 (Rd−1 ) is not sufficient to recover the original random field on S(Rd ).

Example I’. Let Jh be the pseudo differential operators considered in Example I, but here we assume that β > 0 in (3.2) satisfies β ≥ d − 1.

(3.15)

Then, by the discussions given in the former part of Remark 7, for every p ∈ N, : φpJh : can be defined as an S 0 (Rd )-valued random variable. Also, since, in Example I we see that φJh satisfies the reflection positivity in the sense of random variables, by Remark 7 we can conclude that : φpJh : also satisfies the reflection positivity for every p ∈ N. Next, by applying Remark 8 to φJh (and : φpJh : ), we see that the random field having the potential term such that (cf. Remark 8 for the notations used here) exp{−λ } satisfies the reflection positivity, precisely, (2.26) holds by replacing φJ resp. : φ2p J : by φJh resp. : φ2p :. Jh Remark 10 (Generator and essential self adjointness) i) In Example I we can define a time zero field (cf. [N1,2], [Si], [AH-K], [AR]): Let jh be a pseudo differential operator given by (3.1). Let Qh be an L2 space such that Qh ≡ completion (in L2 (Ω; P )) of the linear space spanned by the polynomials Qn such that m=1 < φJhJ , ϕm >, ϕm ∈ Nh , m = 1, . . . , n, n ∈ N. Then, by (3.11), Qh is the d−1 dimensional Gaussian random field with the covariance − 1 h(−∆d−1 + m2 ) 2 . ~ 2 + m2 )β )− 12 , i.e., h(|ξ| ~ 2 + m2 ) = (|ξ| ~ 2 + m2 )β , In particular, if we set jh = (τ 2 + (|ξ| for some β ∈ (0, 2], then Qh becomes the d − 1 dimensional generalized free field with the covariance β (−∆d−1 + m2 )− 2 . ~ 2 + m2 ) = (|ξ| ~ 2 + m2 )2 , i.e., β = 2, then For example, for d = 3 if we set h(|ξ| (3.15) is satisfied and the corresponding time-zero field is the 2-dimensional Nelson’s Euclidean free field [Ne2]. 21

We have to remark that the above discussion gives an interpretation of the present model by means of the stochastic quantization (cf. [AR]), the subsequent remark also has a connection with it. ii) By the same notation Qh , we denote the complexification of the real Hilbert space (for the time-zero field defined above). Then, Qh can be interpreted as a Fock space (cf. [BaSeZh], [Si], [RS], [H], [AFY]). In the following we use the notations that are used in [Si], [RS] (cf. [AFY]) without any explanation. Recall that the symbol h ∈ C ∞ in the pseudo differential operator jh is assumed to 1 satisfy the positivity condition (3.2). Thus, h(−∆d−1 +m2 ) 2 is a pseudo differential 1 operator on S(Rd−1 → C). For the operator h(−∆d−1 + m2 ) 2 consider a dΓ operator on the Hilbert space Qh such that 1 dΓ h(−∆d−1 + m2 ) 2 . (3.16) 1 ˜h given by If we take h(−∆d−1 + m2 ) 2 as an operator in the Hilbert space N ˜h ≡ completion of S(Rd−1 → C) N q with respect to the norm kψkN˜h ≡ (ψ, ψ)N˜h such that Z (ψ, ϕ)N˜h ≡

1 ~ ~ h(|ξ| ~ 2 + m2 ) − 2 dξ, ˆ ξ) ~ ϕ( ˆ ξ) ψ(

ψ, ϕ ∈ S(Rd−1 → C),

Rd−1

then by (3.2)

1 h(−∆d−1 + m2 ) 2 ψ, ψ

˜h N

≥ Cmβ kψk2N˜ . h

(3.17)

1 ˜h , Hence, by taking the Friedrichs-Freudenthal extension of h(−∆d−1 + m2 ) 2 in N this operator becomes a self-adjoint operator that satisfies (3.17). We denote this 1 1 extension by the same notation h(−∆d−1 + m2 ) 2 (in short h 2 ), also we denote its 1 ˜h ( in short D(h 12 )). domain by D h(−∆d−1 + m2 ) 2 ⊂ N 1 ˜h , by (3.17) we can apply Since, h 2 commutes with the complex conjugation in N a theorem on the hypercontractivity (cf. Theorem X.61 of [RS]): On the L2 space Qh the semi group defined by 1 1 Γ e−th 2 = e−tdΓ(h 2 ) ,

t > 0,

(3.18)

1

is hypercontractive. The Hamiltonian dΓ(h 2 ) is essentially self-adjoint on 1 ∞ 2 C dΓ(h ) , the space of analytic vectors (cf. p. 201 of [RS]). Here, by using 1 the same symbol dΓ(h 2 ) we denote its self-adjoint extension (the generator of the 1

semigroup e−tdΓ(h 2 ) ) in (3.18). (For the case where the semigroup is positivity preserving (eg., in the case of a Markov semigroup (cf. the discussions in Remark 9)), the hypercontractivity of the semigroup is equivalent with the property that the generator satisfies the logarithmic Sobolev inequality (cf. [G], [Stro], for its application to the infinite dimensional situations, e.g., [AKRT], [ABRY1], [ABRY2])). Now, in particular, assume that Jh satisfies (3.2), (3.3) with β satisfying (3.15), then by Example I’ we see that for any λ ≥ 0 and any p ∈ N (cf. (2.23) in Remark 8) \ −λ L e ∈ Lq (Ω; P ), (3.19) q≥1

22

where Λd−1 ≡ {~x ∈ Rd−1 | |~x| < L} L

(for some given L ≥ 0).

(3.19) is not completely the same as (2.23), but the time-zero field of φJh is the − 1 Gaussian field with the covariance ( h(−∆d−1 + m2 ) 2 (cf. (3.11)), (3.19) can be proved in the same manner as Remark 8, under the assumption (3.15). Then, by applying Theorem X.58 in [RS], from (3.18) and (3.19) it is possible to see that for each λ ≥ 0, p ∈ N and L ≥ 0 the operator 1

λ ≥ 0, dΓ(h 2 ) + λ · V (φJh ), 1 is essentially self-adjoint on C ∞ dΓ(h 2 ) ∩ D, and bounded below, where V (φJh ) ≡, L

and D ≡ {Y | Y ∈ Qh , V Y ∈ Qh } with Qh defined in i) of this Remark, where Λd−1 is the interior of the d − 1 diL mensional ball having the origin as its center and with radious L ≥ 0 (cf. Remark 8).

Example II. For given d ∈ N and α ∈ (0, 1], let J = J α, and set φJ α = φα . By Proposition 2, for every d ∈ N, α ∈ (0, 1] and p ∈ N, : φpα : does not always become a random variable. For example, for α = 1, the case of Nelson’s Euclidean free field, if d ≤ 2 then : φp1 : is a random variable for every p ∈ N, but if d = 3 then only : φp1 : (p = 1, 2) are random variables and if d ≥ 4 then only φ11 can be defined as a random variable. However it is known (cf. [Klein]) that φα (α ∈ (0, 1]) satisfies the reflection positivity in the sense of random variables (2.6), and it is known as a generalized Euclidean free field. Also, by (1.3) the kernel GJ 2α satisfies the regularity Assumption 1. Combining this and the reflection positivity of φα , we see that φα (α ∈ (0, 1]) satisfies Assumptions 1 and 2, and hence we can apply Theorem 1 to φα (α ∈ (0, 1]). As a consequence, for every d ∈ N, α ∈ (0, 1] and p ∈ N, φα and its Wick power : φpα : (the latter not being necessarily a random variable) satisfy the O-S positivity (2.9). For this example, we can discuss the Osterwalder-Schrader axioms (cf., e.g. [Si], where these axioms are denoted by OS’1-5), and see that : φpα : satisfies all of them: OS 1’) (Temperedness + Analytic continuity) Snp (f ) = Snp (θf ∗ ),

0 Snp ∈ S6= (Rdn ),

n−1

and Snp is a Laplace transform of some Mn ∈ S 0 ((Rd+ ) ), precisely Z Snp (t1 , ~x1 ; . . . ; tn , ~xn ) = M( τ1 , ξ~1 ; . . . ; τn−1 , ξ~n−1 ) n−1 (Rd +)

nn−1 X o × exp iξ~j · (~xj+1 − ~xj ) − τj (tj+1 − tj ) dξ~1 dτ1 · · · dξ~n−1 dτn−1 ; (3.20) j=1

23

OS 2) (Euclidean covariance); OS 3) (Reflection positivity); OS 4) (Symmetry); OS 5) (Cluster property). Since, OS 2) and OS 4) follow from the symmetric structure of Jα , OS 5) follows from the property of moment functions and OS 3) has already been proven above, we have only to show that OS 1’) (i.e. (3.20)) holds. But this can be easily seen by using the explicit formula of the K¨ allen-Lehmann representation for J α given by Corollary 6.4 of [AGW] (cf. also the general K¨allen-Lehmann representation, Theorem IX.34 of [Si]) such that Z ∞ Z √ ~ ~ ρα (dσ 2 ), J 2α (x) = e−τ |t|+ −1ξ·~x dΩσ (τ, ξ) 0

∂Vσ+

where ρα (dσ 2 ) = 2 sin(πα)χ{σ2 >m2 } and ∂Vσ+

dσ 2 , (σ 2 − m2 )α

for α ∈ (0, 1)

ρα (dσ 2 ) = δ{m2 } (σ 2 ) for α = 1, = x ∈ Rd : |t|2 − |~x|2 = σ 2 , t > 0 ,

Ωσ is a measure on ∂Vσ+ such that Z d~x p Ωσ (E) = 2 + |~ σ x|2 pσ (E)

(3.21)

for any measurable set E ∈ ∂Vσ+ ,

and pσ is a homeomorphism of ∂Vσ+ onto Rd−1 such that pσ : (t, x1 , . . . , xd−1 ) 7−→ (x1 , . . . , xd−1 ) = ~x.

By an application of the O.-S. reconstruction theorem to Example II, we have the following. Proposition 11 For any given d ∈ N and p ∈ N, let {SJp α ,n } be the sequence of Schwinger functions considered in Example II. Then the corresponding Wightman functions exist and satisfy all the Wightman axioms. In particular there is an associated Hermitian scalar relativistic quantum field.

Remark 12 The relativistic quantum field defined in Proposition 11 belongs to the Borchers class of the generalized free field, as can be seen similarly as one shows that Wick powers of the free scalar field belong to the Borchers class of this field (cf. [Bo], [Ep], [Rehren]).

Acknowledgement. The second named author is very grateful to Prof.’s Takeyuki Hida, Izumi Ojima, Shigeaki Nagamachi amd Kei-ichi Ito who gave him several important suggestions concerning constructive quantum field theory, through very fruitful discussions on mathematical treatments of this subject. Both authors would like to express their gratitude to Dr. Benedetta Ferrario and Dr. Hanno Gottschalk for the stimulating discussions about the mathematical and physical structure of quantum field theory.

24

The second author also would like to express his gratitude for the financial support by the Ministery of Education, Culture, Sports, Sciences and Technology of Japan, for the research project of “Overseas Research Scholar“. The warm hospitality of the Institut f¨ ur Angewandte Mathematik of Bonn University is gratefully acknowledged, where the main part of this work has been done under the above mentioned research project.

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Zinoviev, Yu.M.: Equivalence of Euclidean and Wightman field theories. Comm. Math. Phys. 174 (1995), no. 1, 1-27.

29

Bestellungen nimmt entgegen: Institut für Angewandte Mathematik der Universität Bonn Sonderforschungsbereich 611 Wegelerstr. 6 D - 53115 Bonn Telefon: Telefax: E-mail:

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Verzeichnis der erschienenen Preprints ab No. 220 220. Otto, Felix; Rump, Tobias; Slepčev, Dejan: Coarsening Rates for a Droplet Model: Rigorous Upper Bounds 221. Gozzi, Fausto; Marinelli, Carlo: Stochastic Optimal Control of Delay Equations Arising in Advertising Models 222. Griebel, Michael; Oeltz, Daniel; Vassilevsky, Panayot: Space-Time Approximation with Sparse Grids 223. Arndt, Marcel; Griebel, Michael; Novák, Václav; Šittner, Petr; Roubíček, Tomáš: Martensitic Transformation in NiMnGa Single Crystals: Numerical Simulation and Experiments 224. DeSimone, Antonio; Knüpfer, Hans; Otto, Felix: 2-d Stability of the Néel Wall 225. Griebel, Michael; Metsch, Bram; Oeltz, Daniel; Schweitzer, Marc Alexancer: Coarse Grid Classification: A Parallel Coarsening Scheme for Algebraic Multigrid Methods 226. De Santis, Emilio; Marinelli, Carlo: Stochastic Games with Infinitely many Interacting Agents 227. Cantero-Álvarez, Rubén; Otto, Felix: The Concertina Pattern: A Supercritical Bifurcation in Ferromagnetic Thin Films 228. Verleye, Bart; Klitz, Margrit; Croce, Roberto; Roose, Dirk; Lomov, Stepan; Verpoest, Ignaas: Computation of Permeability of Textile Reinforcements; erscheint in: Proceedings, Scientific Computation IMACS 2005, Paris (France), July 11-15, 2005 229. Albeverio, Sergio; Pustyl’nikov, Lev; Lokot, Tatiana; Pustylnikov, Roman: New Theoretical and Numerical Results Associated with Dirichlet L-Functions 230. Garcke, Jochen; Griebel, Michael: Semi-Supervised Learning with Sparse Grids; erscheint nd in: Proceedings of 22 Int. Conference on Machine Learning 231. Albeverio, Sergio; Pratsiovytyi, Mykola; Torbin, Grygoriy: Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of their S-Adic Digits 232. Philipowski, Robert: Interacting Diffusions Approximating the Porous Medium Equation and Propagation of Chaos 233. Hahn, Atle: An Analytic Approach to Turaev’s Shadow Invariant

234. Hildebrandt, Stefan; von der Mosel, Heiko: Conformal Representation of Surfaces, and Plateau’s Problem for Cartan Functionals 235. Grunewald, Natalie; Otto, Felix; Reznikoff, Maria G.; Villani, Cédric: A Two-Scale Proof of a Logarithmic Sobolev Inequality 236. Albeverio, Sergio; Hryniv, Rostyslav; Mykytyuk, Yaroslav: Inverse Spectral Problems for Coupled Oscillating Systems: Reconstruction by Three Spectra 237. Albeverio, Sergio; Cebulla, Christof: Müntz Formula and Zero Free Regions for the Riemann Zeta Function 238. Marinelli, Carlo: The Stochastic Goodwill Problem; erscheint in: European Journal of Operational Research 239. Albeverio, Sergio; Lütkebohmert, Eva: The Price of a European Call Option in a BlackScholes-Merton Model is given by an Explicit Summable Asymptotic Series 240. Albeverio, Sergio; Lütkebohmert, Eva: Asian Option Pricing in a Lévy Black-Scholes Setting 241. Albeverio, Sergio; Yoshida, Minoru W.: Multiple Stochastic Integrals Construction of non-Gaussian Reflection Positive Generalized Random Fields

no. 241

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungsgemeinschaft getragenen Sonderforschungsbereiches 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, August 2005 (rev. Fassung Januar 2006)

Multiple stochastic integrals construction of non-Gaussian reflection positive generalized random fields S. Albeverio1),2) , M. W. Yoshida3)∗ 17th Dec. 2005

1) Inst. Angewandte Mathematik, Universit¨ at Bonn, Wegelerstr. 6, D-53115 Bonn (Germany) 2) SFB611; BiBoS; CERFIM, Locarno; Acc. Architettura USI, Mendrisio; Ist. Mathematica, Universit` a di Trento 3) The Univ. Electro commun, Dept. Systems Engineering, 182-8585 Chofu-shi Tokio (Japan) Abstract Multiple stochastic integrals with respect to isonormal Gaussian process on Rd are used for the construction of reflection positive non-Gaussian S 0 (Rd ) valued random variables (random fields). This construction is then extended to a class of Hida distribution that includes the Wick powers of some classes of generalized Euclidean free fields as its elements also to random fields involving potential terms. Under general assumptions, it is proved that the reflection positivity (in the Osterwalder and Schrader sense) are satisfied by these Hida distributions and the random fields. The above newly developed general construction is used to discuss corresponding interesting examples. E.g., it provides a completely different proof of the reflection positivity of the Wick powers of generalized Euclidean free fields, as well as the construction of some new fields (without full Euclidean invariance but satisfying all other axioms, including space rotation invariance) and their Wick powers are the consequences of this probabilistic (and purely Euclidean) procedure.

MSC 2000: 60G15, 60H05, 60H40, 47B25, 81T08, 81T40. Keywords: Multiple stochastic integral,Wick powers, relativistic fields, Euclidean fields, Hida distribution.

0

Introduction

The main considerations performed in this paper are the following two: First, by making use of the multiple stochastic integrals by means of the isonormal Gaussian process on Rd (cf., e.g., Section 4.5 of [H]), we give a new general probabilistic framework to construct non-Gaussian S 0 (Rd ) valued random variables that possess the property of reflection positivity (in the sense of random variables). Where and throughout this paper we denote by d ∈ N the space-time dimension of the random fields and the ∗ e-mail

[email protected] fax +81 424 98 0541.

1

corresponding Euclidean fields, and by N, R, C the spaces of the natural numbers, the real numbers and the complex numbers respectively. This construction of non Gaussian S 0 (Rd ) valued random variables is extended to a class of Hida distributions (generalized random variables in a Kondratiev space), that includes the Wick powers of some class of generalized Euclidean free fields as its elements. We prove that the reflection positivity (in the weaker Osterwalder and Schrader sense (OS positivity), cf. (2.9)) is also satisfied by the elements of Hida distributions in this class (cf. Theorem 2.1). Also we define new reflection positive random fields involving potential terms (without full Eucliden invariance but satisfying all other axioms, cf. Remark 7, 8). Second we discuss interesting examples of interacting quantum fields. Examples provided here explain how the above mentioned general construction is applied. In particular, as an example of reflection positive Hida distributions, we prove by our probabilistic (and, thus, purely Euclidean methods) the reflection positivity of the Wick powers of generalized Euclidean free fields. We would like at this point to recall the fact that the Wick powers of generalized Euclidean free fields need not be S 0 (Rd ) valued random variable. For example, Nelson’s Euclidean free field φN is an S 0 (Rd ) valued random variable, which is a special case of the generalized Euclidean free fields discussed in the present work. The Wick power : φpN : can be defined as an S 0 (Rd ) valued random variable only for the cases where the space time dimensions d ≤ 2 and the powers p ∈ N or d = 3 and p = 1, 2, and for the other cases, e.g., for the higher dimensions d ≥ 4, : φ1N = φN is an S 0 (Rd ) valued random variable but : φpN : (p ≥ 2) are not S 0 (Rd ) valued random variables but actually Hida distributions (cf. Proposition 2, Remark 6), which define forms on the space (S) (cf. [H], [HKPS], [GrotS]). For this reason, the proof of the reflection positivity of the Wick powers of generalized Euclidean free fields can not be accomplished within the framework of the theory of S 0 (Rd )-random fields. Thus, precisely, the organization of this paper and the concepts underlying it will be the following. In section 1, we introduce an isonormal Gaussian process W ≡ {W (h), h ∈ L2 (Rd , λd )} on Rd which is defined on a complete probability space (Ω, F, P ), where λd denotes the Lebesgue measure on Rd . Subsequentially, by means of multiple stochastic integrals with respect to W we construct several S 0 (Rd ) valued random variables defined on the common probability space (Ω, F, P ). Starting from an abstract probability space has the advantage of building a framework for handling all relevant random variables, the probability laws of which correspond to the random fields and phenomena that we want to investigate (cf. [AY], [AFeY], [ABRY1], [ABRY2], [Y4]). This is the spirit of section 1.2 of [H]: ”In constructing or determining a probability space we first clarify the type of random phenomena to be analyzed and the probability structures to be investigated, and then fix Ω, F, P to fit in with these aims”. Following e.g., [AH-K], [AR], [Y1], [Y2], [Y3], we give a construction in this sprit which accomodate the isonormal Gaussian process on Rd (the Gaussian white noise) as well as other generalized white noises (cf. [AW], [AGW], [GrotS]). In this framework various quantities of physical importance can be expressed through suitably chosen random variables. By not fixing a Hilbert space ` a priori, but working with a wide probability space we have a framework which is flexible and permits to develop the analysis of very different phenomena. This probabilistic framework also fits in the spirit of the algebraic approach of quantum field theory where the various non-equilibrium local states are viewed in a common framework (cf. [Oj1] and the review [Oj2]): e.g., the representations of the algebra of the physical quantities characterized and defined through the Gel’fand-Naimark-Segal construction depend on the chosen states, and may be interpreted in the context of the probability theory through the concept of a family

2

of conditional probability measures defined on a given common (sufficiently widely chosen) probability space. Hence, precisely in section 1, for a given pseudo differential operator J on S(Rd ) ~ (τ, ξ) ~ ∈ R × Rd−1 , such that there exist m > 0, C > with a positive symbol j(τ, ξ), 0, α > 0 and ~ ≤ C(|τ |2 + |ξ| ~ 2 + m2 )− α2 , (0.1) 0 < j(τ, ξ) we define an S 0 (Rd ) valued random variable φJ on a complete probability space (Ω, F, P ) as the stochastic integral of the kernel J (using the same notation as for the corresponding pseudo differential operator J) with respect to W , that satisfies the following Z (Jϕ)(x)W (dx), ∀ϕ ∈ S(Rd ). S 0 < φ J , ϕ >S = Rd

~ by making use of multiple stochastic Next, assuming a bound ((1.18) below) for j(τ, ξ), integrals by means of W , we define the p-th Wick power : φpJ : (p ∈ N) of φJ as an S 0 (Rd ) valued random variable, and then specify the support properties of : φpJ : (p ∈ N) in Proposition 2. Such studies on the support properties of S 0 (Rd ) valued random variables are crucially important in the framework of the analysis on an abstract Wiener space (cf., e.g., [AY]), and it is the corresponding consideration for the random field defined by : φpJ : with the consideration on the sample path properties for the stochastic processes, in particular, for Brownian motion process (cf., e.g., section 2.2 of [H]), also in section 3.3 of [H] there is a discussion on the support properties of Gaussian random fields (the discussions given there cover the ~ 2 + m2 )− 12 ). results on φN which is equal to our : φpJ setting p = 1, j = (|τ |2 + |ξ| Corollary 3 will be used to prove the main results in section 2. n p In section 2, the moment functions (Schwinger functions) SJ,n ∈ S 0 ((Rd ) ), n ∈ N, of the S 0 (Rd ) valued random variable : φpJ : are defined under the assumption (1.18) for j: p SJ,n (ϕ1 ⊗ · · · ⊗ ϕn ) ≡ E[< ϕ1 , : φpJ :> · · · < ϕn , : φpJ :>]

ϕr ∈ S(Rd → C),

r = 1, . . . , n.

Then, in Definition 5, by making use of the structure of the above given Schwinger functions, for general J that need not satisfy the bound assumption (1.18) but satisfies n p 0 a regularity assumption (Assumption 1), we define SJ,n ∈ S6= ((Rd ) ), n ∈ N, which p formally corresponds to a ”moment function” of : φJ : (in this case : φpJ : need not be an S 0 (Rd ) valued random variable but only a Hida distribution, and the moments need not exist). Next, in Theorem 1, under the assumption that J satisfies (0.1) with Assumption 1, and that the corresponding S 0 (Rd ) valued random variable φJ satisfies the property p of reflection positivity (2.6), we show in a probabilistic manner that {SJ,n }n∈N defined in Definition 5 satisfies the Osterwalder-Schrader (O-S) positivity (2.9). Remark 7 is a consideration of new random fields satisfying reflection positivity but not satisfying Euclidean invariance. Remark 8 concerns the reflection positive random fields defined by adding a potential terms to the original random fields that satisfy the reflection positivity. Namely, in Remark 8, by assuming both the reflection positivity (2.6) and a new bound (2.20) for j, by making use of the S 0 (Rd ) valued random variable : φ2p J : (p ∈ N), it is shown that for each p ∈ N, Λ ≥ 0 and λ ≥ 0 the random field defined by the following probability measure µJ,p,Λ,λ ≡ µ on S 0 (Rd ) satisfies the property of reflection positivity (cf. [GRS], [AH-K], [AR], and for

3

another direction in this concern, [AY]): Z < ϕ1 , ω 0 > · · · < ϕn , ω 0 > µ(dω 0 ) S 0 (Rd )

≡

2p 1 E[e−λ < ϕ1 , φJ > · · · < ϕn , φJ >], ZΛ 2p

where ZΛ = E[e−λ ], amd IBΛ is the indicator function with BΛ ≡ {x ∈ Rd | |x| < Λ}. The considerations on such random fields in the contexts of Euclidean quantum field theory is new. In section 3, we examine some examples. An example having the correspondence to Remark 7 is Example I, where we study the random fields φJh , for Jh having the following form: 1 ∂2 (0.2) Jh = {− 2 + h(−∆d−1 + m2 )}− 2 , ∂t where h ∈ C ∞ (R+ → R+ ) is some given function such that h(x) ≥ C|x|β ,

∀x ∈ R+ ,

for some β > 0, C > 0, 2

(0.3) 2

∂ ∂ and ∆d−1 is the d − 1 dimensional Laplacian: ∆d−1 ≡ ∂x In 2 + · · · + ∂x 2. 1 d−1 Example I’, as a special case of Example I, we study the random fields φJh on Rd having Jh given by (0.2) which satisfy (0.3) with β ≥ d − 1. Remark 8 applies to such φpJh . In Example II, we discuss the case where

J = Jα ≡ (−

α ∂2 − ∆d−1 + m2 )− 2 , ∂t

α ∈ (0, 1].

In fact the generalized Euclidean free field, φJα and its Wick power : φpJα : characterized by Jα , are considered. By applying Theorem 2.1, the O-S positivity of the generalized Euclidean fields corresponding to them is proven and the corresponding Schwinger functions admit an analytic continuation. Before closing Introduction, for an adequate understanding of the concept of probabilistic proof of the reflection positivity of non-Gaussian random variables (and generalized random variables) indexed by S(Rd ), including Wick powers of the generalized Euclidean free fields as its special case, we should review the existing results and discussions having a connection with this subject: In the relativistic quantum field theory, generalized free fields were introduced by Greenberg [Gre] as natural generalizations of free fields satisfying all the axioms (resp. breaking part of them in a simple way), see, e.g. [Jo], [BogLogT], [Wi1,2] and [StWi] (p.105). Generalized free fields also occurs in a natural way in the discussion of certain ”structual questions” and ”triviality results”, see e.g. [FernFS], [DeA]. Wick powers of free fields have been studied extensively, see e.g. [StWi], [Bo], [LToll], [AFeY]. It is also natural to study in a similar way Wick powers of generalized free fields. This has been sketched in [Wi1,2]. A use of them for the construction of models of 2 space-time dimensional quantum fields has been done in [Ha1], [Ha2]. In [Klein], [Okabe], [SH] the O-S positivity of Euclidean generalized free fields is discussed. We have to point out that following the lines of [Wi1] and [Wi2], there is an alternative proof of the reflection positivity of the generalized Euclidean fields : φpJα : discussed in Example II: The ”Wightman functions” given by the analytic continuations of the {SJpα ,n }n∈N , can namely be identified with the known Wightman functions corresponding to the Wick powers of generalized free fields in relativistic quantum 4

field theory. In general starting from relativistic Hermitian scalar fields satisfying the Wightman axims and applying the fact that they imply the Osterwalder-Schrader axioms” (cf. [OS], [OS2], [Si], [Gl], [He], [OcaOtt]) one gets, in particular, the reflection positivity of the corresponding Euclidean fields. Thus, by the analytic continuation, once the Wick powers of generalized Euclidean free fields are identified with the Wick powers of generalized free fields in relativistic quantum field theory, then one can proceed using the fact that Wick powers of generalized free fields are Wightman fields (cf. [Wi3], [Br¨ u], [SS1,2] [Streater], [D¨ uR], [Licht1,2]), to prove the reflection positivity in Example II. However, as compared with this ”‘indirect proof”’, our probabilistic proof has the advantage of being direct (i.e. it is performed without analytic continuation from the Euclidean to the relativistic domain) and involves stochastic analytic estimates of interest by themselves. Our Theorem 1, Remarks 7 and 8 are in this way fully in accord with the Euclidean strategy in constructive quantum field theory, where one tries to verify the axioms for Euclidean field theory through arguments in the Euclidean regions. In this concern we may find a similarity between the present consideration and the ones which lead Zinoviev in [Z 1995] to modify the OS 1’ axiom (cf. (3.20)) to get a complete mathematical equivalence between Euclidean and relativistic axioms (see also [NaMu1,2,3]); ([OS2] does not have a complete equivalence in this sense as first pointed out in [Gl]).

1

Preliminaries

Let S(Rd ) be the Schwartz space of rapidly decreasing test functions equipped with the usual topology by which it is a Fr´echet nuclear space. Let S 0 (Rd ) be the topological dual space of S(Rd ). We denote the Fourier and Fourier inverse transform of a function ϕ ∈ S(Rd ) re√ R − −1x·ξ −1 −d 2 e ϕ(x)dx, F −1 [ϕ](ξ) = spectively by F[ϕ] and F [ϕ]: F[ϕ](ξ) = (2π) Rd √ R −1x·ξ d −d ϕ(x)dx for ϕ ∈ S(R ). We sometimes use the notation F[ϕ] = ϕ. ˆ (2π) 2 Rd e d By J we denote a pseudo differential operator on S(R ), for which we assume the following: There exist positive numbers m > 0, C > 0 and α > 0 such that α

~ ≤ C(|(τ, ξ)| ~ 2 + m2 )− 2 , 0 < j(τ, ξ)

~ ∈ R × Rd−1 , ∀(τ, ξ)

(1.1)

where ~ ≡ |(τ, ξ)|

q 2 , τ 2 + ξ12 + · · · ξd−1

τ ∈ R,

ξ~ ≡ (ξ1 , . . . , ξd−1 ) ∈ Rd−1 ,

and j ∈ S(Rd → R) is the symbol of the pseudo differential operator J: (Jϕ)(t, ~x) = F −1 (j ϕ)(t, ˆ ~x),

for

ϕ ∈ S(Rd ), (t, ~x) ∈ R × Rd−1 .

By the same notation J we denote the kernel corresponding to the pseudo differential operator as follows: Z (Jϕ)(t, ~x) = J(t − t0 , x1 − x01 , . . . , xd−1 − x0d−1 )ϕ(t0 , x01 , . . . , x0d−1 )dt0 dx01 · · · dx0d−1 , Rd

with the kernel defined by J(t, ~x) ≡ (F −1 j)(t, ~x). Throughout this paper we often use the notations ~ ∈ R × Rd−1 . ξ ≡ (τ, ξ)

x ≡ (t, ~x) ∈ R × Rd−1 , 5

(1.2)

α

We denote, in particular, the pseudo differential operator (−∆d +m2 )− 2 by J m,α , α > 0 (if there is no ambiguity, we often omit m > 0 and simply denote this by J α ), where ∆d denote the d-dimensional Laplace operator. By (1.2) we see that J m,α (x) ≥ 0 has the following regularlity (cf. [NaM3] and (3.13) in Example I): There exist some constants C1 , C2 > 0 (depending on m, α) such that ( C1 |x|−d+α for |x| < 1 m,α J (x) ≤ (1.3) C1 e−C2 |x| for |x| ≥ 1. Let η1 ∈ C0∞ (Rd ) be such that ( 0 ≤ η1 (x) ≤ 1

and

η1 (x) =

1

|x| ≤ 1

0

|x| ≥ 2.

(1.4)

and let ηk (x) = η1 ( xk ) ∈ C0∞ (Rd ), k = 1, 2, . . . . Also define ρ ∈ C0∞ (Rd ) as follows: 1 C exp(− ) |x| < 1 , (1.5) ρ(x) = 1 − |x|2 0 |x| ≥ 1 R where the constant C is taken to satisfy Rd ρ(x)dx = 1. Define ρk (x) = k d ρ(kx). (1.6) For the kernel of a pseudo differential operator J, of which symbol satisfies (1.1), we define a sequence of smooth functions Jk ∈ S(Rd ), k = 1, 2, . . . as follows: Z Jk (x) = J(y)ρk (x − y)dy = (ρk ∗ J)(x), x ∈ Rd . (1.7) Rd

Also, by the p ∈ N times multiplication of Jk we define, for yi ∈ Rd , i = 1, . . . , p, FkJ,p (x; y1 , . . . , yp ) = (ηk (x))p Jk (x − y1 ) · · · Jk (x − yp ).

(1.8)

Then the function FkJ,p is symmetric in the last p variables (y1 , . . . , yp ) and p+1

FkJ,p ∈ S((Rd )

FkJ,p (x; y1 , . . . , yp ) = 0

),

for |x| ≥ 2k.

(1.9)

For later convenience let us recall the following properties: ξ ρˆk (ξ) = ρˆ( ), k

|ˆ ρ(ξ)| ≤ 1,

ρˆ(0) = 1

(by (1.5)).

(1.10)

We also recall that ρˆk (ξ) converges to 1 uniformly on compact sets: For any M < ∞ and any > 0 there exists an N < ∞ such that 0 ≤ 1 − ρˆk (ξ) < , for any ξ such that |ξ| ≤ M and any k ≥ N . Now, suppose that on a complete probability space (Ω, F, P ) we are given an isonormal Gaussian process W = {W (h), h ∈ L2 (Rd ; λd )}, where λd denotes the Lebesgue measure on Rd (cf., e.g., [H], [Nu], [BaSeZh], [Y4]): W is a centered Gaussian family of random variables such that Z E[W (h)W (g)] = h(x) g(x)λd (dx), h, g ∈ L2 (Rd ; λd ). (1.11) Rd

We write

Z Wω (h) =

h(y)Wω (dy), Rd

6

ω∈Ω

with Wω (·) a Gaussian generalized random variable (in the notation of Hida calculus ˙ (y)dy cf. [HKPS]). for the Gaussian white noise Wω (dy) is often written as W For the pseudo differential operator J satisfying (1.1) and for each p ∈ N, we define a sequence of random variable :k φpJ,ω :, k ∈ N by means of the multiple stochastic integral (multiple Wiener integral) as follows (cf., e.g., [Nu], [AY]): Z p :k φJ,ω : (x) = FkJ,p (x; y1 , . . . , yp )Wω (dy1 ) · · · Wω (dyp ) x ∈ Rd . (1.12) (Rd )p

In the sequel, for p = 1 we use the simpler notation such that :k φ1J,ω : (x) = k φJ,ω (x). By this definition, from (1.9) we can take :k φpJ,ω : as a C0 (Rd → R) ≡ C0 (Rd )valued random variable. In fact, there exists a bounded open set Dk = {x | |x| < 2k} ⊂ Rd such that :k φpJ,ω : (x) = 0

for x ∈ Rd \ Dk

∀ω ∈ Ω,

and moreover, by the Kolmogorov’s continuity criterion the stochastic process {:k φpJ,ω : (x)}x∈Rd admits a continuous modification (cf., e.g.,[Nu]), which we also denote by :k φpJ,ω : (x): :k φpJ,ω : (·) ∈ C0 (Rd → R)

∀ω ∈ Ω.

(1.13)

For each a, b, d > 0, we define a linear subspace Bda,b of S 0 (Rd ) as follows: a

b

Bda,b = {(|x|2 + 1) 4 (−∆d + 1)+ 2 f : f ∈ L2 (Rd ; λd )},

(1.14)

where λ denotes the Lebesgue measure on R. Then Bda,b becomes a separable Hilbert space with the scalar product Z b b u, v ∈ Bda,b . < u|v >= J 1,a ((1 + |x|2 )− 4 u(x)) J 1,a ((1 + |x|2 )− 4 v(x))dx, Rd

(1.15) The following proposition 1 is easy to be provem in the same manner as in [Y4], where the case J = J α was considerd. Proposition 1 Let J be a pseudo differential operator satisfying (1.1), and :k φpJ,ω : be the random variable defined by (1.12). Then, for each k ∈ N and r ≥ 1 there exists Mk,r and the following holds: Z Z r p (1.16) :k φJ,ω : (x) dxP (dω) < Mk,r . Ω

Rd

Also for each k and l let U k,l (x1 , . . . , xl ) ≡ E (:k φpJ,· : (x1 )) · · · (:k φpJ,· : (xl )) , then U k,l ∈ C0 ((Rd )l → R).

(1.17)

Since (1.13) holds and C0 (Rd → R) ⊂ Bda,b (a, b, d > 0), we can consider the limit k → ∞ for :k φpJ,ω : (·) ∈ C0 (Rd → R) in Bda,b . As a consequence, we have the following Proposition 2, which can be proven in the same manner as in [Y4], where the case of J = J α was proved.

7

Proposition 2 integer p satisfy

Suppose that J satisfies (1.1) for some α > 0. Let the positive min{1,

2α 2a } + p × min{1, } > p, d d

(1.18)

for some a ∈ R, then the sequence of S 0 (Rd )-valued random variables {:k φpJ,ω :}k∈N has a limit in the space of Bda,b (b > d) valued random variables, hence the limit, denoted by : φpJ,ω :, is a Bda,b -valued random variable. Moreover the following hold: i) Z k :k φpJ,ω : − :m φpJ,ω : k2B a,b P (dω) = 0;

lim

k,m→∞

d

Ω

ii) There exists a P -null set N , a subsequence {:kj φpJ,ω :} of {:k φpJ,ω :} and a Bda,b -valued random variable : φpJ,ω : such that lim k :kj φpJ,ω : − : φpJ,ω : kB a,b = 0,

kj →∞

∀ω ∈ Ω \ N ;

d

For each ϕ ∈ S(Rd ) there exists a P -null set Nϕ and

iii)

S 0 ,S = lp,ω (ϕ)

∀ω ∈ Ω \ Nϕ ,

lim k S 0 ,S − S 0 ,S kL2 (Ω;P ) = 0,

k→∞

where Z lp,ω (ϕ)

= (Rd )p

Z (

ϕ(x) J(x − y1 ) · · · J(x − yp ) dx)

Rd

× Wω (dy1 ) · · · Wω (dyp );

(1.19)

iv) In particular for p = 1, (1.18) alwys holds for J satisfying (1.1), and a Bda,b valued random variable φJ,ω ≡: φ1J,ω : can be defined by ii). By Proposition 2-iv), since φJ,ω is an S 0 (precisely a Bda,b -) valued random variable, by Proposition 2-iii) the process ηk (x) < φJ,ω , Jk (x − ·) >S 0 ,S is a continuous process with parameter x ∈ Rd . On the other hand, by (1.13) since k φJ,ω (x) = R η (x)Jk (x − y)Wω (dy) is a continuous process with parameter x ∈ Rd , the two Rd k continuous processes (or C0 (Rd → R)-valued random variables) ηk (x) < φJ,ω , Jk (x − ·) > and k φJ,ω (x) are equivalent. We state this fact in Corollary 3-i) ( Corollary 3-ii) is a well known fact of the Wiener chaos decomposition). Corollary 3 Suppose that J satisfies (1.1). Let φJ ≡: φ1J : be the S 0 -valued random variable defined by Proposition 2-iv). Then the following statements are true: i) The continuous processes ηk (x) < Jk (x − ·), φJ,ω > and R 2 d k φJ,ω (x) = Rd ηk (x)Jk (x − y)Wω (dy) in L (Ω; P ) with parameter x ∈ R are equivalent in the following sense: there exists a P -null set N such that for all ω ∈ Ω \ N , k φJ,ω (x)

ii)

=< ηk (x)ρk (x − ·), φJ,ω >S,S 0

∀x ∈ Rd .

(1.20)

For any positive integer p p

:k φpJ,ω

[2] X

1 : (x) = p! m!(p − 2m)! m=0

m p−2m 1 − bk (x) , k φJ,ω (x) 2 8

(1.21)

where bk (x) is a function depending on J such that Z 2 2 bk (x) = ηk (x) Jk (x − z) dz. Rd

2

Reflection positive fields

We start with the consideration of the moment functions corresponding to the p-th Wick power of φJ in case where : φpJ : is a random variable. By Proposition 2, for J satisfying (1.18) for a power p under consideration, : φpJ : can be defined as an S 0 -valued random variable, and we can define the moment functions: for each p ∈ N (satisfying (1.18)) and n ∈ N, let spJ,n (ϕ1 ⊗ · · · ⊗ ϕn ) ≡ E[< ϕ1 , : φpJ :> · · · < ϕn , : φpJ :>], ϕr ∈ S(Rd → C),

(2.1)

r = 1, . . . n.

From (1.11) and (1.19), by setting 2 GJ = kernel of the pseudo differential operator with the symbol j(ξ) , then spn can be expressed by the explicit formula: p spJ,n (ϕ1 ⊗ · · · ⊗ ϕn ) =< SJ,n , ϕ1 ⊗ · · · ⊗ ϕn >S,S 0 ,

(2.2)

where p SJ,n (x1 , . . . , xn ) =

X

r(l)σn,p (GJ (xj − xi ) : l)

for np = even number

l∈Ln,p 0

otherwise,

(2.3) Ln,p is the totality of sequences l = (l1,2 , . . . , l1,n ; l2,3 , . . . , l2,n ; . . . ; ln−1,n ) such that li,j = 0, 1, . . . , p n X

l1,j = p,

l1,2 +

j=2

n X

l2,j = p,

l1,3 + l2,3 +

j=3

...,

k−1 X

li,k +

i=1

n X

j3,j = p, . . .

j=4 n X

lk,j = p, . . . ,

n−1 X

li,n = p.

(2.4)

i=1

j=k+1

and for each l ∈ Ln,p , σn,p (xj,i : l) = σn,p (x2,1 , . . . , xn,1 ; x3,2 . . . , xn,2 ; . . . ; xn,n−1 : l) is a monomial of xj,i ∈ R (i = 1, 2, . . . , n − 1, j = i + 1, . . . , n) defined by σn,p (xj,i : l) =

n n Y Y

(xj,i )li,j ,

i=1 j=i+1

with

also

r(l) =

Y

rij (l)

(2.5)

i 0 |GJ (x − x0 )| ≤ c(|~x − x~0 |l + 1) × (|t − t0 |r + |t − t0 |−r ), ∀x = (t, ~x) ∈ R × Rd−1 ,

x0 = (t0 , x~0 ) ∈ R × Rd−1 .

Note that Assumption 1 is satisfied by GJ α , (α > 0), by (1.3). Definition 5 Suppose that J satisfies (1.1) and Assumption 1. For each positive p integer p ∈ N we define SJ,n (n = 1, 2, . . . ) by (2.3). We call the continuous linear p dn functional SJ,n on S6= (R ) the n-th Schwinger function associated with : φpJ :. For p n = 0, we set SJ,0 ≡ 1. We note that under the regularity Assumption 1 for GJ , the fact that the above p 0 defined Schwinger functions SJ,n are elements of S6= (Rdn ) is obvious. Remark 6 (Hida distribution) In the framework of Hida calculus, Proposition 2 and Definition 5 are expressed as follows (here we use the notations of Hida calculus without any explanation, for these notations and notions cf. [GrotS] and [KLS]): d Suppose that J satisfies (1.1) for some α ∈ (0, d2 ), then for p ∈ N such that p < d−2α p (cf. (1.18)), the random variable defined in Proposition 2 is expressed by , where (only here) ω is the d-dimensional Gaussian white noise and Z (F J,p ϕ)(y1 , . . . , yp ) ≡ ϕ(x) J(x − y1 ) · · · J(x − yp ) dx, ϕ ∈ S Rd

10

(the ω in this context should of course not be confused with the generic point in Ω in d , are defined as Hida the rest of the paper). Also, for p ≥ d−2α distributions in (S)−1 , which are the continuous functionals on (S)1 such that >(S)−1 ,(S)1 Z = p! (F J,p ϕ)(y1 , . . . , yp )ψ(y1 , . . . , yp )dy1 · · · dyp , Rdp

for ψ ∈ S((Rd )p ), where I(ψ) =. Moreover (cf. [KLS], [GrotS]), it is possible to define as an element of (S)−1 for each x ∈ Rd , where δ{x} is the Dirac point measure at x, and the following holds Z ϕ(x)dx =, ϕ ∈ S. Rd

Namely, the notion : φpJ : (and the Shwinger functions corresponding to it) given in Definition 5 has the expression by means of a Hida distribution: . Definition 5 says that generally speaking, an element of (S)−1 , , only defines a form on (S), however, for J satisfying (1.1) and Assumption 1, the expectation E( · · · ) 0 has a meaning and defines an element of S6= (Rdn ).

To state the reflection positivity of fields on S(Rd ) we need the notion of the time reflection operator. The time reflection operator θ is defined by θ(t, ~x) = (−t, ~x), as a map from Rd into Rd . We also denote by θ its induced action on elements n fn ∈ S((Rd+ ) ) such that θfn (x1 , · · · , xn ) = fn (θ−1 x1 , · · · , θ−1 xn ). Assumption 2 Assume that J satisfies (1.1) for some α > 0. Suppose also that a,b the Bd -valued random variable φJ,ω on (Ω, F, P ) defined by Proposition 2-iv) satisfies the property of reflection positivity in the sense of random variables (cf., e.g., Definition 5 of [AGW]): For anr ∈ C, ϕn,r ∈ S(Rd+ → C), r = 1, . . . , Nn , l = 1, . . . , n, n = 1, . . . , N l (Nn , N ∈ N) and c ∈ C, n X Nn N X o ¯n,r a¯nr < θϕ¯n,r ¯ E n , φJ > · · · < θϕ1 , φJ > + c n=1 r=1

×

Nn N X nX

o n,r anr < ϕn,r , φ > · · · < ϕ , φ > + c ≥ 0. J J n 1

(2.6)

n=1 r=1

Examples of J by which Assumption 2 is satisfied are given in the next section (cf. Example I, Remark 9 and Example II). We remark that for φJ,ω satisfying Assumption 2, the corresponding Schwinger functions {s1J,n }n∈N∪{0} satisy the following O-S positivity not only for fn ∈ S+ (Rdn ), but also for fn ∈ S((Rd+ )n ): N X N X

n

1 Sn+m (θfn∗ ⊗ fm ) ≥ 0 ∀f0 ∈ C, fn ∈ S((Rd+ ) ),

n=0 m=0

11

n = 1, . . . , N,

(2.7)

where fn∗ (x1 , . . . , xn ) = fn (xn , . . . , x1 ). Now, we restrict the space of test functions to S6= (Rdn ), and extend the results on reflection positivity to the case of Wick powers. Theorem 2.1 Assume that for a given J, Assumption 1 and Assumption 2 are p satisfied. Let {SJ,n }n∈N be the sequence of Schwinger functions defined by Definition 5. Then p 0 SJ,n ∈ S6= (Rdn ) (2.8) p and {SJ,n } satisfies the O-S positivity: N X N X

p SJ,n+m (θfn∗ ⊗ fm ) ≥ 0

for all

f0 ∈ C, fn ∈ S+ (Rdn ), n = 1, . . . , N.

n=0 m=0

(2.9) In particular, if α > 0 in (1.1) satisfies α ≥ d2 , then for every p ∈ N, : φpJ : is an S 0 (Rd )-valued random variable and satisfies the reflection positivity in the sense of random variables, namely it satisfies (2.6) with the replacement of φJ by : φpJ . Proof. (2.8) is a direct consequence of the regularity Assumption 1 for GJ . In order to prove (2.9), it suffices to show that N N X X

p (θfn∗ ⊗ fm ) ≥ 0 SJ,n+m

(2.10)

n=0 m=0

for fn of the special form fn = such that

PNn

r=1

n,r dn anr ϕn,r 1 ⊗ · · · ⊗ ϕn ∈ S+ (R ), n = 0, . . . , N ,

ϕn,r ∈ C0∞ (Rd → R) ∩ S(Rd+ → R), l

l = 1, . . . , n,

r = 1, . . . , Nn ,

(2.11)

(the general case follows by a continuity and density argument). We prove (2.10) only for N = 2, N1 = 0 and N2 = 1 (for other N and Nn the proof can be carried out similarly, and it will be briefly explained after finishing the proof for these general cases). For each k ∈ N define GJ,k (xj , xi ) by Z GJ,k (xj , xi ) = FkJ,2 (y; xj , xi )dy, (2.12) Rd

where FkJ,2 is defined by (1.8). Using GJ,k (xj , xi ) for given p we define p SJ,n,(k) (x1 , . . . , xn ) with the replacement of GJ (xj −xi ) by GJ,k (xj , xi ) in (2.3). Then for f = ϕ1 ⊗ ϕ2 ∈ S+ (R2d → R) such that ϕi ∈ C0∞ (Rd → R) ∩ S(Rd+ → R), i = 1, 2, by Fubini’s lemma and then Lebesgue’s convergence theorem we see that, for ∀c ∈ R, p p p SJ,4 (θf ⊗ f ) + cSJ,2 (θf ) + cSJ,2 (f ) + c2 n o p p p = lim SJ,4,(k) (θf ⊗ f ) + cSJ,2,(k) (θf ) + cSJ,2,(k) (f ) + c2 . k→∞

12

(2.13)

On the other hand by (1.11), (1.12), (1.16) and Fubini’s lemma p p p SJ,4,(k) (θf ⊗ f ) + cSJ,2,(k) (θf ) + cSJ,2,(k) (f ) + c2 = E {< θϕ2 , :k φpJ :>< θϕ1 , :k φpJ :> +c}{< ϕ1 , :k φpJ :>< ϕ2 , :k φpJ :> +c} n Z Z o =E ϕ2 (θx) :k φpJ : (x)dx ϕ1 (θx) :k φpJ : (x)dx + c Rd Rd Z n Z o × ϕ1 (x) :k φpJ : (x)dx ϕ2 (x) :k φpJ : (x)dx + c d Rd Z R = ϕ1 (x)ϕ1 (x0 )ϕ2 (y)ϕ2 (y0 ) R4d h i ×E :k φpJ : (x) :k φpJ : (θx0 ) :k φpJ : (y) :k φpJ : (θy0 ) dxdx0 dydy0 Z h i +c ϕ1 (x)ϕ2 (y)E :k φpJ : (x) :k φpJ : (y) dxdy 2d ZR h i +c ϕ1 (x)ϕ2 (y)E :k φpJ : (θx) :k φpJ : (θy) dxdy + c2 . (2.14) R2d

By (2.13) and (2.14), in order to show that the LHS of (2.13) is not less than 0 it suffices to prove that the RHS of (2.14) is not less than 0 when k is large enough. For above ϕ1 and ϕ2 let δ be such that δ ≡ min d(supp[ϕ1 ], ∂Rd+ ), d(supp[ϕ1 ], supp[ϕ2 ]) where d(A, B) represents the (Euclidean) distance between two sets A and B, and M ∂Rd+ is the boundary of Rd+ . For each M ∈ N we take {AM j }j=1,...,M and {Bj }j=1,...,M as coverings of supp[ϕ1 ] respectively supp[ϕ2 ], such that d d(AM j , ∂R+ ) > δ −

1 , M

SM

j=1

M [

M d(AM j , Bj ) > δ −

AM j and

SM

AM j ⊃ supp[ϕ1 ],

j=1

1 , M

∀j, ∀j 0 = 1, . . . , M ;

BjM are compact,

M 0 AM j ∩ Aj 0 = ∅ (j 6= j ),

j=1

|supp[ϕ1 ]| − 1 /M ≤ |AM j | ≤ |supp[ϕ1 ]| + 1 /M, M [

BjM ⊃ supp[ϕ2 ],

∀j = 1, . . . , M ;

BjM ∩ BjM0 = ∅ (j 6= j 0 ),

j=1

|supp[ϕ2 ]| − 1 /M ≤ |BjM | ≤ |supp[ϕ2 ]| + 1 /M, ∞ [ M \

∞ [ M \

AM j = supp[ϕ1 ],

M =1 j=1

∀j = 1, . . . , M ;

BjM = supp[ϕ2 ],

M =1 j=1

where |A| denots the volume (evaluated by Lebesgue measure) of a set A. M M For each M and j we denote some points in the partition AM j and Bj by xj 4

and yjM respectively. By (1.17) we know that U k,4 ∈ C0 ((Rd ) → R) and U k,2 ∈ 2

C0 ((Rd ) → R), and the RHS of (2.14) is a pointwise limit of Riemann sums, i.e.: The RHS of (2.14) = lim Ek (M ) + c2 , M →∞

13

(2.15)

where Ek (M )

=

M X M X M X M X i=1

× U + c

k,4

i0 =1

j=1

M M ϕ2 (yjM0 )ϕ1 (xM i0 )ϕ1 (xi )ϕ2 (yj )

j 0 =1

M M M M M M (θyjM0 , θxM i0 , xi , yj )|Ai0 ||Bj 0 ||Ai ||Bj |

M X M X

M k,2 M M ϕ1 (xM (xi , yjM )|AM i )ϕ2 (yj )U i ||Bj |

i=1 j=1

+ c

M X M X

M k,2 M M ϕ1 (xM (θyjM0 , θxM i0 )ϕ2 (yj 0 )U i0 )|Ai0 ||Bj 0 |.

i=10 j 0 =1 1 < δ and supp[ϕ1 ] ∪ supp[ϕ2 ] ⊂ It can be seen that if k and M satisfy k1 + M {x | |x| < k}, then Ek (M ) + c2 ≥ 0. (2.16)

For this purpose, by Proposition 1 and again by Fubini’s lemma we write Ek (M ) + c2 as follows: Ek (M ) + c2 n X M X M o p M M M =E ϕ2 (yjM0 )(:k φpJ : (θyiM0 ))ϕ1 (xM )(: φ : (θx ))|A ||B | + c 0 0 0 0 k J i i i j j 0 =1 i0 =1 M X M nX

×

o p p M M M M M ϕ1 (xM )(: φ : (x ))ϕ (y )(: φ : (y ))|A ||B | + c . (2.17) k J 2 j k J i i j i j

i=1 j=1

To show that (2.16) holds we crucially use (1.20) and (1.21). We give the detailed proof of (2.16) for p = 2 (for p ≥ 3 the proof of (2.16) is similar). Since k is taken to satisfy supp[ϕ1 ] ∪ supp[ϕ2 ] ⊂ {x | |x| < k}, by (1.4) and the definition of ηk we have M M M ηk (xM i ) = ηk (θxi ) = ηk (yj ) = ηk (θyj ) = 1,

i, j = 1, . . . M.

By this for i, j = 1, . . . , M , bk (xM i )

=

bk (θxM i )

=

bk (yjM )

=

bk (θyjM )

Z = bk ≡

2 Jk (· − z) dz.

Rd

Hence, for p = 2 by (1.20) and (1.21) we have M X M X

2 M M 2 M M M ϕ1 (xM i )(:k φJ : (xi ))ϕ2 (yj )(:k φJ : (yj ))|Ai ||Bj | + c

i=1 j=1

=

M X M X

n o M M M M 2 ϕ1 (xM − bk (xM k φJ (xi ) i )ϕ2 (yj )|Ai | |Bj | i )

i=1 j=1

n o 2 × k φJ (yjM ) − bk (yjM ) + c =

M X M X

aij < fi , φJ >< fi , φJ >< gj , φJ >< gj , φJ >

i=1 j=1

+

M X

ai < fi , φJ >< fi , φJ > +

i=1

M X j=1

14

a0j < gj , φJ >< gj , φJ > +a0 , (2.18)

where M M M aij = ϕ1 (xM i )ϕ2 (yj )|Ai | |Bj |,

fi (·) = ρk (xM i − ·),

M X M ai = −bk ϕ1 (xM ϕ2 (yjM )|BjM |}, i )|Ai |{

gj (·) = ρk (yjM − ·),

M X M a0j = −bk ϕ2 (yjM )|BjM |{ ϕ1 (xM i )|Ai |},

j=1

i=1 2

a0 = c + (bk )

M X M X

M M M ϕ1 (xM i )ϕ2 (yj )|Ai | |Bj |.

i=1 j=1

Also since ρ(θx − y) = ρ(x − θy) we have M X M X

2 M M M ϕ2 (yjM )(:k φ2J : (θyjM ))ϕ1 (xM i )(:k φJ : (θxi ))|Ai ||Bj | + c

j=1 i=1

= a0 +

M X M X

aij < θgj , φJ >< θgj , φJ >< θfi , φJ >< θfi , φJ >

i=1 j=1

+

M X

a0j < θgj , φJ >< θgj , φJ > +

j=1

M X

ai < θfi , φJ >< θfi , φJ > . (2.19)

i=1

For ϕ1 ⊗ϕ2 ∈ S+ (R2d → R) such that ϕi ∈ C0∞ (Rd → R)∩S(Rd+ → R), i = 1, 2, d 1 1 M since ρk (xM i −·), ρk (yj −·) ∈ S(R+ → R) for every i, j = 1, . . . , M when k + M < δ, supp[ϕ1 ] ∪ supp[ϕ2 ] ⊂ {x | |x| < k} (cf. (1.6)), by (2.18) and (2.19) from (2.6) the RHS of (2.17) is non-negative for such k and M , i.e. (2.16) holds (note that φJ,ω satisfies the reflection positivity in the sense (2.6) of random variables); by (2.16) and (2.15) the RHS of (2.14) is non-negative for such k; by this the LHS of (2.13) is nonnegative and hence (2.10) has been proved for N = 2, N1 = 0, N2 = 1 and p = 2. The other cases can be proved in a similar way: RHS of the formula (2.18) corresponding to (cf. (2.14)) < ϕ1 , :k φpJ :> · · · < ϕn , :k φpJ :> +c becomes a polymomial of < f, φJ >, f ∈ C0∞ (Rd → R) ∩ S(Rd+ → R), with the power np, and the non-negativity of (2.17) corresponding to the general cases are proved through the same arguments performed above. Remark 7 (Modification of the case α ≥ d2 ) Suppose that J satisfies the following: There exist m > 0, C > 0, β > 0 and γ > 12 such that β≥

d−1 , 2γ − 1

and n o− γ2 ~ ≤ C |τ |2 + (|ξ| ~ 2 + m2 )β 0 ≤ j(τ, ξ) ,

~ ∈ R × Rd−1 . ∀(τ, ξ)

(2.20)

By the same argument as for Proposition 2, under the condition (2.20) we easily see that for every p ∈ N, : φpJ : can be defined as an S 0 (Rd ) valued random variable taking values in the same region (sub space of S 0 (Rd )) in which : φpJ α : with α ≥ d2 takes 15

values. As a consequence, for : φpJ α : (p ∈ N) with J satisfying (2.20), Proposition 2 holds for every p ∈ N. Also, by (1.20) and (1.21), for ϕ ∈ S(Rd+ → R) the random variable < ϕ, : T p φJ,ω :>S,S 0 ∈ q≥1 Lq (Ω; P ) is σ{< ϕ, φJ,ω >S,S 0 |ϕ ∈ S(Rd+ → R)} measurable, where σ{< ϕ, φJ,ω >S,S 0 |ϕ ∈ S(Rd+ → R)} is the sub σ-field of F generated by the random variables < ϕ, φJ,ω >S,S 0 , ϕ ∈ S(Rd+ → R). Since, if we assume in addition that φJ satisfies (2.6), the reflection positivity in the sense of random variables, then by the Lebesgue’s convergence theorem (on the probability space (Ω, F, P )) every σ{< ϕ, φJ,ω >S,S 0 |ϕ ∈ S(Rd+ → R)} measurable L2 (Ω; P ) random variable satisfies the reflection positivity in the sense of random variables, we see that for every p ∈ N and n ∈ N the random variable < ϕ1 , : φpJ,ω :> · · · < ϕn , : φpJ,ω :>, ϕi ∈ S(Rd+ → R), i = 1, . . . , n, satisfies the reflection positivity , namely it satisfies (2.6) by replacing φJ by : φpJ :. In such a way, if : φpJ : is an S 0 -valued random variable, then the proof of reflection positivity becomes simpler, but for : φpJ : that is not a random variable but only a Hida distribution, then in order to get the result of O-S positivity (the O-S axioms [OS1,2] expression of reflection positivity) we should add a regularity assumption for GJ (cf. Assumption 1, cf. also [NaMu1,2,3] for more singular ”Shwinger functions” corresponding to hyperfunctions ). Remark 8 (Adding potential terms) For A ∈ B(Rd ), let χA denote the ind dicator function, where B(R ) is the Borel σ-field of Rd . Assume that J satisfies (2.20) or (1.1) with α ≥ d2 . Then Proposition 2-iii) holds not only for ϕ ∈ S but also for χA , precisely, for every p ∈ N it is possible to show that Z lim k :k φpJ : (x)dx − lp,ω (χA )kL2 (Ω;P ) = 0 ∀A ∈ B(Rd ), (2.21) k→∞

A

where lp,ω (χA ) is the random variable defined by (1.19) by setting ϕ = χA (χA is not an element of S but (1.19) is well defined by (1.11)). Because of (2.21), we will denote lp,ω (χA ) by . Now, for L > 0, let ΛoL,+ ≡ {(t, ~x) ∈ R × Rd−1 : |(t, ~x)| < L,

t > 0},

ΛoL,− ≡ {(t, ~x) ∈ R × Rd−1 : |(t, ~x)| < L,

t < 0},

ΛL,+ ≡ {(t, ~x) ∈ R × R

d−1

: |(t, ~x)| < L,

t ≥ 0},

ΛL,− ≡ {(t, ~x) ∈ R × Rd−1 : |(t, ~x)| < L,

t ≤ 0},

ΛL ≡ {(t, ~x) ∈ R × R

d−1

: |(t, ~x)| < L},

Then, through a direct calculation by using (1.11) and (1.19), for any p ∈ N we have k − kL2 (Ω;P ) = 0, k − kL2 (Ω;P ) = 0, Hence, for P − a.e. ω we see that =, = . (2.22) Next, recall that for the Gaussian random fields defined by stochastic integrals and multiple stochastic integrals of Jϕ (ϕ ∈ S), such that J satisfies (1.1) with α ≥ d2 , 16

the following evaluation holds (cf. Th. 1.22 of [Si], for applications see, e.g., [AY], [AFY]): m (q ≥ 2), kΦkLq (Ω;P ) ≤ (q − 1) 2 kΦkL2 (Ω;P ) , where, for fl ∈ S(Rd → R), l = 1, . . . , m, m ∈ N, Z Φ= (Jf1 )(x1 ) · · · (Jfm )(xm )W (dx1 ) · · · W (dxm ). (Rd )m

The above is also true for J satisfying (2.20). By making use of this bound, through the same arguments as in Theorem V.7 in [Si], for J satisfying (2.20) or (1.1) with α ≥ d2 , we have for any λ ≥ 0 and p ∈ N, o exp{−λ },

o exp{−λ },

exp{−λ } ∈

\

Lq (Ω; P ).

(2.23)

q≥1

By (2.22) and (2.23) we have \ 2p o o Lq (Ω; P ) 3 exp{−λ } × exp{−λ } q≥1

=

exp{−λ },

P − a.s..

(2.24)

d o Since, exp{−λ } is σ{< ϕ, φJ >S,S 0 : ϕ ∈ S(R+ → R)} measurable, and 2p o o exp{−λ } = exp{−λ }

(2.25)

is σ{< ϕ, φJ >S,S 0 : ϕ ∈ S(Rd− → R)} measurable, where Rd− ≡ {x ∈ Rd | x = (t, ~x) ∈ R×Rd−1 , t < 0}, if we assume in addition that φJ satisfies Assumption 2 (the reflection positivity in the sense of random variables), then by the result of Remark 7 ( the fact that σ{< ϕ, φJ,ω >S,S 0 |ϕ ∈ S(Rd+ → R)} measurable L2 (Ω, P ) random variable is reflection positive) , from (2.24) and (2.25) we see that the following holds: For anr ∈ C, ϕn,r ∈ S(Rd+ → C), r = 1, . . . , Nn , l = 1, . . . , n, n = 1, . . . , N l (Nn , N ∈ N) and c ∈ C, Nn N X nX o 2p ¯n,r E e−λ a¯nr < θϕ¯n,r ¯ n , φJ > · · · < θϕ1 , φJ > + c n=1 r=1

×

Nn N X nX

anr

···

+ c /ZΛ ≥ 0,

n=1 r=1

where

2p

ZΛ = E[e−λ ].

3

Examples

In this section, we apply Theorem 1 and Remark 8 to three examples.

17

(2.26)

Example I. For given (space time) dimension d ∈ N and mass m > 0, suppose that the symbol of the pseudo differential operator J is given by 1 ~ ∈ R × Rd−1 , ~ ≡ τ 2 + h(|ξ| ~ 2 + m2 ) − 2 , (τ, ξ) (3.1) jh (τ, ξ) where h ∈ C ∞ (R → R+ ) is a given function that satisfies h(x) ≥ C|x|β ,

∀x ∈ R,

(3.2)

for some β > 0, C > 0. In this section, these pseudo differential operators with symbols having the form given by (3.1) will be denoted by Jh . Then Jh is expressed by Jh = −

− 1 ∂2 + h(−∆d−1 + m2 ) 2 , 2 ∂t

where ∆d−1 is the d − 1 dimensional Laplacian:

∆d−1 =

(3.3) ∂2 ∂x21

+ ··· +

∂2 . ∂x2d−1

By

(3.1) and (3.2), since jh satisfies (1.1), from Proposition 2-iv), φJh is defined as an S 0 (Rd )-valued random variable. In order to apply Theorem 1 to φJh and its Wick powers, we have to show that 1◦ ) φJh satisfies the reflection positivity in the sense of random variable; 2◦ ) the integral kernel GJh satisfies the regularity Assumption 1. By modifying a standard discussion for Euclidean invariant random fields introduced by [N1,2] (cf. also [Si]), we can show 1◦ ) in the following way Let Hh ≡ completion of S(Rd → R) with respect to the norm Z 12 ~ ϕ(τ, ~ 2 dτ dξ~ , ϕ ∈ S(Rd → R). kϕkHh ≡ (jh (τ, ξ) ˆ ξ)) R×Rd−1

Then Hh ⊂ S 0 (Rd → R) is a Hilbert space equipped with the inner product given by (where we use the complex Hilbert space notation) Z ~ 2 ϕ(τ, ~ ψ(τ, ˆ ξ)dτ ~ dξ. ~ (ϕ, ψ)Hh = (jh (τ, ξ)) ˆ ξ) (3.4) Rd

Similar to the discussion for (2.21), < φJh , ϕ > can be defined as an L2 (Ω; P ) random variable for any ϕ ∈ Hh . Let Mh ≡ ϕ ∈ Hh | supp[ϕ] ⊂ {(t, ~x) ∈ Rd | t ≤ 0} , Nh ≡ ϕ ∈ Hh | supp[ϕ] ⊂ {(t, ~x) ∈ Rd | t = 0} . Now, take ϕ ∈ Hh such that supp[ϕ] ⊂ {(t, ~x) ∈ Rd | t > 0},

(3.5)

and let ϕ˜ be the orthogonal projection of ϕ onto Mh . Then (cf. Theorem 5 of [N2]) < ψ, (−

∂2 ∂2 2 −1 +h(−∆ +m )) ϕ ˜ >=< ψ, (− +h(−∆d−1 +m2 ))−1 ϕ >, ∀ψ ∈ Mh . d−1 ∂t2 ∂t2

In the above equation, by setting ψ ∈ C0∞ (Rd− ) ⊂ Mh with supp[ψ] ⊂ {(t, ~x)| t < 0}, we see that (−

∂2 ∂2 2 −1 + h(−∆ + m )) ϕ ˜ = (− + h(−∆d−1 + m2 ))−1 ϕ d−1 ∂t2 ∂t2

18

on {(t, ~x)| t < 0}. (3.6)

2

∂ 2 But the operator − ∂t 2 +h(−∆d−1 +m ) keeps the support unchanged on the time axis ∂2 2 −1 of the functions. Thus, by (3.5) and (3.6) (by operating (− ∂t 2 + h(−∆d−1 + m )) both side of (3.6)) we see that

supp[ϕ] ˜ ⊂ {(t, ~x) | t = 0} i.e.

ϕ˜ ∈ Nh 0

for

ϕ ∈ Hh

satisfying (3.5).

By this it follows that the S (R → R)-valued random variable φJh is an S 0 (Rd−1 → R)-valued Markov process with the parameter t ∈ R, namely for ϕ ∈ Hh satisfying (3.5) the following holds: d

E[< φJh , ϕ > | σ(Mh )] = E[< φJh , ϕ > | σ(Nh )],

P − a.s.,

(3.7)

where σ(Mh ) and σ(Nh ) are the σ-fields generated by the random variables {< φJh , ψ > | ψ ∈ Mh } and {< φJh , ψ > | ψ ∈ Nh } respectively. Next, following the proof of Theorem 1 in [N1] (cf., also, the proof of Proposition III.4-(a) of [Si], for more general situations cf. Theorem 2.14 of [Mizohata]) we use the fact that ψ ∈ Nh admits a representation such that ψ(t, ~x) =

n X

gm (~x)δt=0 ,

for suitable gm ∈ S 0 (Rd−1 → R), m = 1, . . . , n,

n ∈ N,

m=1

(3.8) where δt=0 is the Dirac point measure at t = 0. From (3.8), by using (1.11), (1.19) and (3.1), for ϕ ∈ Hh satisfying (3.5), passing through a direct calculation we see that E[< φJh , ϕ >< φJh , ψ >] = E[< φJh , θϕ >< φJh , θψ >],

∀ψ ∈ Nh ,

(3.9)

where θ is the time reflection operator defined in the previous section. Hence, for ϕ ∈ Hh satisfying (3.5) we see that the following holds (”reflection property” p.101 in [N1]): E[< φJh , ϕ > | σ(Nh )] = E[< φJh , θϕ > | σ(Nh )],

P − a.s..

2

(3.10)

∂ 2 −1 need not satisfy the We should remark that the operator (− ∂t 2 + h(−∆d−1 + m )) maximum principle, which is used in the proof of Theorem 1 of [N1] where the case of ∂2 2 −1 (− ∂t (the Nelson’s Euclidean free field) is discussed. For random 2 − ∆d−1 + m ) fields that are not necessarily Euclidean invariant to prove the ”reflection property” we only need the fact that the expression (3.8) holds for ϕ ∈ Nh (i.e. we only need the regularity on the time axis, cf. Remark 9, below). By (3.7) and (3.10), we have 1◦ ) immediately: In fact, for ϕ ∈ Hh satisfying (3.5) we get

E[< φJh , ϕ >< φJh , θϕ >] h i = E E[< φJh , ϕ > | σ(Mh )] < φJh , θϕ > h i = E E[< φJh , ϕ > | σ(Nh )] < φJh , θϕ > h i = E E E[< φJh , ϕ > | σ(Nh )] < φJh , θϕ > σ(Nh ) h i = E E[< φJh , ϕ > | σ(Nh )]E[< φJh , θϕ > | σ(Nh )] ≥ 0. 19

Since, the above calculation is valid for the replacement of < φJh , ϕ >< φJh , θϕ > by the random variable inside of the expectation of the LHS of (2.6), we have thus proven 1◦ ). To show that 2◦ ) holds, we remark that ~ (t, ~x) GJh (t, ~x) ≡ F −1 j 2 (τ, ξ) 1 Z √ −1 ~ 2 + m2 ) −1 dτ (~x) √ e −1t·τ τ 2 + h(|ξ| = Fd−1 2π R r √ π 1 2 ~2 −1 q e− h(|ξ| +m )|t| (~x), (3.11) = Fd−1 2 ~ 2 + m2 ) h(|ξ| −1 where Fd−1 denotes the Fourier inverse transform with respect to the variable ξ~ ∈ d−1 R . For β > 0 given in (3.2), define

r = r(d, β) ≡ min{r0 ∈ N | βr0 ≥ d}.

(3.12)

Since, √ 2 ~2 e− h(|ξ| +m )|t|

1 q

∞ n 1 X i−1 1 ~ 2 + m2 ) 2 |t|n ~ 2 + m2 ) 2 h(|ξ| h(|ξ| n! n=0

=

h

≤

|t|−r

~ 2 + m2 ) h(|ξ|

r+1 1 ~ 2 + m2 ) − 2 , h(|ξ| r!

by (3.11) we see that |GJh (t, ~x)| ≤ C|t|−r , where C=

1 r!

Z

∀~x ∈ Rd−1 ,

r+1 ~ 2 + m2 ) − 2 dξ~ < ∞. h(|ξ|

(3.13)

(3.14)

Rd−1

Hence, GJh satisfies Assumption 1 for r > 0, C > 0 given by (3.12) and (3.14) resp., and l = 0. This then proves 2◦ ). As a consequence, by applying Theorem 1 we see that (2.9) holds for φJh and its Wick powers, that are not necessarily random variables. Remark 9 (Reflection property) that jh has the form

As a generalization of Example I, suppose

k ~ ≡ τ 2 + h(|ξ| ~ 2 + m2 ) − 2 , jh (τ, ξ)

for some k ∈ N,

where h ∈ C ∞ (R → R+ ) satisfies (3.2) as before. Then, the discussion for the derivation of (3.7) is also valid for this general case (i.e., for any k ∈ N): for ϕ ∈ Hh satisfying (3.5) we have E[< φJh , ϕ > | σ(Mh )] = E[< φJh , ϕ > | σ(Nh )],

P − a.s..

(3.70 )

However, for k ≥ 2 we do not have the expression (3.8) for ψ ∈ Nh any more, namely 0 ψ ∈ Nh may have a component such that g(~x)δt=0 , where g(~x) ∈ S 0 (Rd−1 → R) and 0 δt=0 is the derivative of the Dirac point measure δt=0 , hence (3.9) need not hold and we do not have the ”reflection property” (3.10) (for these discussions cf. [N2]). 20

As a consequence, even though the Markov property (3.7’) holds, we can not get the reflection positivity for φJh with Jh given above (k ≥ 2). In terms of probability theory, this discussion can be expressed as follows: For the case where k ≥ 2, the S 0 (Rd )-valued random variable φJh satisfies a generalized Markovian property property with respect to σ(Nh ) and it is also an S 0 (Rd−1 )-valued stochastic process with the parameter t ∈ R, but it is not an S 0 (Rd−1 )-valued (symmetric) Markov process, because the σ-field generated by the random variables with test functions ψ having the form (3.8) is exactly a sub σ-field of σ(Nh ). Namely the information available through S 0 (Rd−1 ) is not sufficient to recover the original random field on S(Rd ).

Example I’. Let Jh be the pseudo differential operators considered in Example I, but here we assume that β > 0 in (3.2) satisfies β ≥ d − 1.

(3.15)

Then, by the discussions given in the former part of Remark 7, for every p ∈ N, : φpJh : can be defined as an S 0 (Rd )-valued random variable. Also, since, in Example I we see that φJh satisfies the reflection positivity in the sense of random variables, by Remark 7 we can conclude that : φpJh : also satisfies the reflection positivity for every p ∈ N. Next, by applying Remark 8 to φJh (and : φpJh : ), we see that the random field having the potential term such that (cf. Remark 8 for the notations used here) exp{−λ } satisfies the reflection positivity, precisely, (2.26) holds by replacing φJ resp. : φ2p J : by φJh resp. : φ2p :. Jh Remark 10 (Generator and essential self adjointness) i) In Example I we can define a time zero field (cf. [N1,2], [Si], [AH-K], [AR]): Let jh be a pseudo differential operator given by (3.1). Let Qh be an L2 space such that Qh ≡ completion (in L2 (Ω; P )) of the linear space spanned by the polynomials Qn such that m=1 < φJhJ , ϕm >, ϕm ∈ Nh , m = 1, . . . , n, n ∈ N. Then, by (3.11), Qh is the d−1 dimensional Gaussian random field with the covariance − 1 h(−∆d−1 + m2 ) 2 . ~ 2 + m2 )β )− 12 , i.e., h(|ξ| ~ 2 + m2 ) = (|ξ| ~ 2 + m2 )β , In particular, if we set jh = (τ 2 + (|ξ| for some β ∈ (0, 2], then Qh becomes the d − 1 dimensional generalized free field with the covariance β (−∆d−1 + m2 )− 2 . ~ 2 + m2 ) = (|ξ| ~ 2 + m2 )2 , i.e., β = 2, then For example, for d = 3 if we set h(|ξ| (3.15) is satisfied and the corresponding time-zero field is the 2-dimensional Nelson’s Euclidean free field [Ne2]. 21

We have to remark that the above discussion gives an interpretation of the present model by means of the stochastic quantization (cf. [AR]), the subsequent remark also has a connection with it. ii) By the same notation Qh , we denote the complexification of the real Hilbert space (for the time-zero field defined above). Then, Qh can be interpreted as a Fock space (cf. [BaSeZh], [Si], [RS], [H], [AFY]). In the following we use the notations that are used in [Si], [RS] (cf. [AFY]) without any explanation. Recall that the symbol h ∈ C ∞ in the pseudo differential operator jh is assumed to 1 satisfy the positivity condition (3.2). Thus, h(−∆d−1 +m2 ) 2 is a pseudo differential 1 operator on S(Rd−1 → C). For the operator h(−∆d−1 + m2 ) 2 consider a dΓ operator on the Hilbert space Qh such that 1 dΓ h(−∆d−1 + m2 ) 2 . (3.16) 1 ˜h given by If we take h(−∆d−1 + m2 ) 2 as an operator in the Hilbert space N ˜h ≡ completion of S(Rd−1 → C) N q with respect to the norm kψkN˜h ≡ (ψ, ψ)N˜h such that Z (ψ, ϕ)N˜h ≡

1 ~ ~ h(|ξ| ~ 2 + m2 ) − 2 dξ, ˆ ξ) ~ ϕ( ˆ ξ) ψ(

ψ, ϕ ∈ S(Rd−1 → C),

Rd−1

then by (3.2)

1 h(−∆d−1 + m2 ) 2 ψ, ψ

˜h N

≥ Cmβ kψk2N˜ . h

(3.17)

1 ˜h , Hence, by taking the Friedrichs-Freudenthal extension of h(−∆d−1 + m2 ) 2 in N this operator becomes a self-adjoint operator that satisfies (3.17). We denote this 1 1 extension by the same notation h(−∆d−1 + m2 ) 2 (in short h 2 ), also we denote its 1 ˜h ( in short D(h 12 )). domain by D h(−∆d−1 + m2 ) 2 ⊂ N 1 ˜h , by (3.17) we can apply Since, h 2 commutes with the complex conjugation in N a theorem on the hypercontractivity (cf. Theorem X.61 of [RS]): On the L2 space Qh the semi group defined by 1 1 Γ e−th 2 = e−tdΓ(h 2 ) ,

t > 0,

(3.18)

1

is hypercontractive. The Hamiltonian dΓ(h 2 ) is essentially self-adjoint on 1 ∞ 2 C dΓ(h ) , the space of analytic vectors (cf. p. 201 of [RS]). Here, by using 1 the same symbol dΓ(h 2 ) we denote its self-adjoint extension (the generator of the 1

semigroup e−tdΓ(h 2 ) ) in (3.18). (For the case where the semigroup is positivity preserving (eg., in the case of a Markov semigroup (cf. the discussions in Remark 9)), the hypercontractivity of the semigroup is equivalent with the property that the generator satisfies the logarithmic Sobolev inequality (cf. [G], [Stro], for its application to the infinite dimensional situations, e.g., [AKRT], [ABRY1], [ABRY2])). Now, in particular, assume that Jh satisfies (3.2), (3.3) with β satisfying (3.15), then by Example I’ we see that for any λ ≥ 0 and any p ∈ N (cf. (2.23) in Remark 8) \ −λ L e ∈ Lq (Ω; P ), (3.19) q≥1

22

where Λd−1 ≡ {~x ∈ Rd−1 | |~x| < L} L

(for some given L ≥ 0).

(3.19) is not completely the same as (2.23), but the time-zero field of φJh is the − 1 Gaussian field with the covariance ( h(−∆d−1 + m2 ) 2 (cf. (3.11)), (3.19) can be proved in the same manner as Remark 8, under the assumption (3.15). Then, by applying Theorem X.58 in [RS], from (3.18) and (3.19) it is possible to see that for each λ ≥ 0, p ∈ N and L ≥ 0 the operator 1

λ ≥ 0, dΓ(h 2 ) + λ · V (φJh ), 1 is essentially self-adjoint on C ∞ dΓ(h 2 ) ∩ D, and bounded below, where V (φJh ) ≡, L

and D ≡ {Y | Y ∈ Qh , V Y ∈ Qh } with Qh defined in i) of this Remark, where Λd−1 is the interior of the d − 1 diL mensional ball having the origin as its center and with radious L ≥ 0 (cf. Remark 8).

Example II. For given d ∈ N and α ∈ (0, 1], let J = J α, and set φJ α = φα . By Proposition 2, for every d ∈ N, α ∈ (0, 1] and p ∈ N, : φpα : does not always become a random variable. For example, for α = 1, the case of Nelson’s Euclidean free field, if d ≤ 2 then : φp1 : is a random variable for every p ∈ N, but if d = 3 then only : φp1 : (p = 1, 2) are random variables and if d ≥ 4 then only φ11 can be defined as a random variable. However it is known (cf. [Klein]) that φα (α ∈ (0, 1]) satisfies the reflection positivity in the sense of random variables (2.6), and it is known as a generalized Euclidean free field. Also, by (1.3) the kernel GJ 2α satisfies the regularity Assumption 1. Combining this and the reflection positivity of φα , we see that φα (α ∈ (0, 1]) satisfies Assumptions 1 and 2, and hence we can apply Theorem 1 to φα (α ∈ (0, 1]). As a consequence, for every d ∈ N, α ∈ (0, 1] and p ∈ N, φα and its Wick power : φpα : (the latter not being necessarily a random variable) satisfy the O-S positivity (2.9). For this example, we can discuss the Osterwalder-Schrader axioms (cf., e.g. [Si], where these axioms are denoted by OS’1-5), and see that : φpα : satisfies all of them: OS 1’) (Temperedness + Analytic continuity) Snp (f ) = Snp (θf ∗ ),

0 Snp ∈ S6= (Rdn ),

n−1

and Snp is a Laplace transform of some Mn ∈ S 0 ((Rd+ ) ), precisely Z Snp (t1 , ~x1 ; . . . ; tn , ~xn ) = M( τ1 , ξ~1 ; . . . ; τn−1 , ξ~n−1 ) n−1 (Rd +)

nn−1 X o × exp iξ~j · (~xj+1 − ~xj ) − τj (tj+1 − tj ) dξ~1 dτ1 · · · dξ~n−1 dτn−1 ; (3.20) j=1

23

OS 2) (Euclidean covariance); OS 3) (Reflection positivity); OS 4) (Symmetry); OS 5) (Cluster property). Since, OS 2) and OS 4) follow from the symmetric structure of Jα , OS 5) follows from the property of moment functions and OS 3) has already been proven above, we have only to show that OS 1’) (i.e. (3.20)) holds. But this can be easily seen by using the explicit formula of the K¨ allen-Lehmann representation for J α given by Corollary 6.4 of [AGW] (cf. also the general K¨allen-Lehmann representation, Theorem IX.34 of [Si]) such that Z ∞ Z √ ~ ~ ρα (dσ 2 ), J 2α (x) = e−τ |t|+ −1ξ·~x dΩσ (τ, ξ) 0

∂Vσ+

where ρα (dσ 2 ) = 2 sin(πα)χ{σ2 >m2 } and ∂Vσ+

dσ 2 , (σ 2 − m2 )α

for α ∈ (0, 1)

ρα (dσ 2 ) = δ{m2 } (σ 2 ) for α = 1, = x ∈ Rd : |t|2 − |~x|2 = σ 2 , t > 0 ,

Ωσ is a measure on ∂Vσ+ such that Z d~x p Ωσ (E) = 2 + |~ σ x|2 pσ (E)

(3.21)

for any measurable set E ∈ ∂Vσ+ ,

and pσ is a homeomorphism of ∂Vσ+ onto Rd−1 such that pσ : (t, x1 , . . . , xd−1 ) 7−→ (x1 , . . . , xd−1 ) = ~x.

By an application of the O.-S. reconstruction theorem to Example II, we have the following. Proposition 11 For any given d ∈ N and p ∈ N, let {SJp α ,n } be the sequence of Schwinger functions considered in Example II. Then the corresponding Wightman functions exist and satisfy all the Wightman axioms. In particular there is an associated Hermitian scalar relativistic quantum field.

Remark 12 The relativistic quantum field defined in Proposition 11 belongs to the Borchers class of the generalized free field, as can be seen similarly as one shows that Wick powers of the free scalar field belong to the Borchers class of this field (cf. [Bo], [Ep], [Rehren]).

Acknowledgement. The second named author is very grateful to Prof.’s Takeyuki Hida, Izumi Ojima, Shigeaki Nagamachi amd Kei-ichi Ito who gave him several important suggestions concerning constructive quantum field theory, through very fruitful discussions on mathematical treatments of this subject. Both authors would like to express their gratitude to Dr. Benedetta Ferrario and Dr. Hanno Gottschalk for the stimulating discussions about the mathematical and physical structure of quantum field theory.

24

The second author also would like to express his gratitude for the financial support by the Ministery of Education, Culture, Sports, Sciences and Technology of Japan, for the research project of “Overseas Research Scholar“. The warm hospitality of the Institut f¨ ur Angewandte Mathematik of Bonn University is gratefully acknowledged, where the main part of this work has been done under the above mentioned research project.

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29

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Verzeichnis der erschienenen Preprints ab No. 220 220. Otto, Felix; Rump, Tobias; Slepčev, Dejan: Coarsening Rates for a Droplet Model: Rigorous Upper Bounds 221. Gozzi, Fausto; Marinelli, Carlo: Stochastic Optimal Control of Delay Equations Arising in Advertising Models 222. Griebel, Michael; Oeltz, Daniel; Vassilevsky, Panayot: Space-Time Approximation with Sparse Grids 223. Arndt, Marcel; Griebel, Michael; Novák, Václav; Šittner, Petr; Roubíček, Tomáš: Martensitic Transformation in NiMnGa Single Crystals: Numerical Simulation and Experiments 224. DeSimone, Antonio; Knüpfer, Hans; Otto, Felix: 2-d Stability of the Néel Wall 225. Griebel, Michael; Metsch, Bram; Oeltz, Daniel; Schweitzer, Marc Alexancer: Coarse Grid Classification: A Parallel Coarsening Scheme for Algebraic Multigrid Methods 226. De Santis, Emilio; Marinelli, Carlo: Stochastic Games with Infinitely many Interacting Agents 227. Cantero-Álvarez, Rubén; Otto, Felix: The Concertina Pattern: A Supercritical Bifurcation in Ferromagnetic Thin Films 228. Verleye, Bart; Klitz, Margrit; Croce, Roberto; Roose, Dirk; Lomov, Stepan; Verpoest, Ignaas: Computation of Permeability of Textile Reinforcements; erscheint in: Proceedings, Scientific Computation IMACS 2005, Paris (France), July 11-15, 2005 229. Albeverio, Sergio; Pustyl’nikov, Lev; Lokot, Tatiana; Pustylnikov, Roman: New Theoretical and Numerical Results Associated with Dirichlet L-Functions 230. Garcke, Jochen; Griebel, Michael: Semi-Supervised Learning with Sparse Grids; erscheint nd in: Proceedings of 22 Int. Conference on Machine Learning 231. Albeverio, Sergio; Pratsiovytyi, Mykola; Torbin, Grygoriy: Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of their S-Adic Digits 232. Philipowski, Robert: Interacting Diffusions Approximating the Porous Medium Equation and Propagation of Chaos 233. Hahn, Atle: An Analytic Approach to Turaev’s Shadow Invariant

234. Hildebrandt, Stefan; von der Mosel, Heiko: Conformal Representation of Surfaces, and Plateau’s Problem for Cartan Functionals 235. Grunewald, Natalie; Otto, Felix; Reznikoff, Maria G.; Villani, Cédric: A Two-Scale Proof of a Logarithmic Sobolev Inequality 236. Albeverio, Sergio; Hryniv, Rostyslav; Mykytyuk, Yaroslav: Inverse Spectral Problems for Coupled Oscillating Systems: Reconstruction by Three Spectra 237. Albeverio, Sergio; Cebulla, Christof: Müntz Formula and Zero Free Regions for the Riemann Zeta Function 238. Marinelli, Carlo: The Stochastic Goodwill Problem; erscheint in: European Journal of Operational Research 239. Albeverio, Sergio; Lütkebohmert, Eva: The Price of a European Call Option in a BlackScholes-Merton Model is given by an Explicit Summable Asymptotic Series 240. Albeverio, Sergio; Lütkebohmert, Eva: Asian Option Pricing in a Lévy Black-Scholes Setting 241. Albeverio, Sergio; Yoshida, Minoru W.: Multiple Stochastic Integrals Construction of non-Gaussian Reflection Positive Generalized Random Fields