FEYNMAN-KAC SEMIGROUPS, MARTINGALES AND WAVE

J. Korean Math. Soc. 38 (2001), No. 2, pp. 227–274

FEYNMAN-KAC SEMIGROUPS, MARTINGALES AND WAVE OPERATORS

Jan A. Van Casteren Abstract. In this paper we intend to discuss the following topics: (1) Notation, generalities, Markov processes. The close relationship between (generators of) Markov processes and the martingale problem is exhibited. A link between the Korovkin property and generators of Feller semigroups is established. (2) Feynman-Kac semigroups: 0-order regular perturbations, pinned Markov measures. A basic representation via distributions of Markov processes is depicted. (3) Dirichlet semigroups: 0-order singular perturbations, harmonic functions, multiplicative functionals. Here a representation theorem of solutions to the heat equation is depicted in terms of the distributions of the underlying Markov process and a suitable stopping time. (4) Sets of finite capacity, wave operators, and related results. In this section a number of results are presented concerning the completeness of scattering systems (and its spectral consequences). (5) Some (abstract) problems related to Neumann semigroups: 1st order perturbations. In this section some rather abstract problems are presented, which lie on the borderline between first order perturbations together with their boundary limits (Neumann type boundary conditions and) and reflected Markov processes.

Received July 14, 1999. Revised March 30, 2000. 2000 Mathematics Subject Classification: 47D06, 47D02, 60J25, 60J45, 81Q10. Key words and phrases: Feynman-Kac semigroups, martingales, multiplicative functionals, capacity theory. The author is obliged to the University of Antwerp (UIA) and the Flemish Fund for Scientific Research (FWO) for their material support. He also sincerely thanks J. C. Zambrini for some valuable references and ideas: see Theorem 15. He is also indebted to B. Boufoussi for some references (like [18, [45]), and discussions on problems related to Neumann semigroups. The hospitality at Yonsei University, Department of Mathematics, where the author was a guest of Kun Soo Chang, is gratefully acknowledged. The author is also thankful to an anonymous referee for corrections and useful comments.

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1. Notation, generalities, Markov processes The following notation will be used throughout this paper. By E we mean a locally compact second countable Hausdorff space; E 4 = E ∪{4} is the one-point compactification of E, or, if E is itself compact, then 4 is an isolated point of E 4 . The state space E (E 4 ) is supplied with its Borel field E (E4 ). The sample space Ω := D [0, ∞] , E 4 , called Skorohod space, which is a polish space for a suitable metric, is given by Ω = {ω : [0, ∞] → E 4, ω cadlag on [0, ζ), ω(s)=4, t ≥ s ⇒ ω(t) = 4}; The symbol ζ stands for the life time ζ(ω) = inf{s > 0 : ω(s) = 4} of the sample path ω. The state variables X(t) : Ω → E 4 are defined by X(t)(ω) = ω(t). The σ-fields Ft , t ≥ 0, constitute a filtration; in principle they are given by Ft = σ(X(s) : 0 ≤ s ≤ t). Their interpretation is that of information from the past. The σ-field generated by {X(s) : s ≥ t} is interpreted as the information from the future. The σ-field {X(t)∈B : B ∈ E} is interpreted as the present information. The family (Ft )t≥0 is called the history of the process. By F we mean F = σ(X(s) : s ≥ 0). The (time) translation operators are defined by [ϑt (ω)] (s) = ω(s + t). By definition, a function f : E → C belongs to the space C0 (E), if it is continuous and if for every ε > 0 there exists a compact subset K = Kε of E such that |f (x)| < ε for x ∈ / K. Definition 1. A family {S(t) : t ≥ 0} of operators defined on L∞ (E) is a Feller semigroup on C0 (E) if it possesses the following properties: (i) It leaves C0 (E) invariant: S(t)C0 (E) ⊆ C0 (E) for t ≥ 0; (ii) It is a semigroup: S(s + t) = S(s) ◦ S(t) for all s, t ≥ 0, and S(0) = I; (iii) It consists of contraction operators: kS(t)f k∞ ≤ kf k∞ for all t ≥ 0 and for all f ∈ C0 (E); (iv) It is positivity preserving: f ≥ 0, f ∈ C0 (E), implies S(t)f ≥ 0; (v) It is continuous for t = 0: limt↓0 [S(t)f ] (x) = f (x), for all f ∈ C0 (E) and for all x ∈ E. In the presence of (iii) and (ii), property (v) is equivalent to: (v0) limt↓0 kS(t)f − f k∞ = 0 for all f ∈ C0 (E). So that a Feller semigroup is in fact strongly continuous in the sense that lim||S(s)f − s→t

Feynman-Kac semigroups, martingales and wave operators

229

S(t)f ||∞ = 0, for every f ∈ C0 (E). A strongly continuous semigroup {S(t) : t ≥ 0} is called a Feller semigroup if it possesses the following positivity property: for all f ∈ C0 (E), for which 0 ≤ f ≤ 1, and for all t ≥ 0, the inequality 0 ≤ S(t)f ≤ 1 is true. Parts of Theorem 2 are proved in [48]. Theorem 2. (a) (Blumenthal and Getoor [5]) Let {S(t) : t ≥} be a Feller semigroup in C0 (E). Then there exists a strong Markov process (in fact a Hunt process) (1.1)

{(Ω, F, Px ) , (X(t) : t ≥ 0) , (ϑt : t ≥ 0) , (E, E)} ,

(1.2)

[S(t)f ] (x) = Ex [f (X(t))] , f ∈ C0 (E), t ≥ 0.

such that

Moreover this Markov process is normal (i.e. Px [X(0) = x] = 1), is right continuous (i.e. limt↓s X(t) = X(s), Px -almost surely), possesses left limits in E on its life time (i.e. limt↑s X(t) exists in E, whenever ζ > s), and is quasi-left-continuous (i.e. if (Tn : n ∈ N) is an increasing sequence of (Ft )-stopping times, X(Tn ) converges Px -almost surely to X(T ) on the event {T < ∞}, where T = supn∈N Tn ). (b) Conversely, let {(Ω, F, Px ) , (X(t) : t ≥ 0) , (ϑt : t ≥ 0) , (E, E)} be a strong Markov process which is normal, right continuous, and possesses left limits in E on its life time. Put [S(t)f ] (x) = Ex [f (X(t))] for f a bounded Borel function, t ≥ 0, x ∈ E. Suppose that S(t)f belongs to C0 (E) for f belonging to C0 (E), t ≥ 0. Then {S(t) : t ≥ 0} is a Feller semigroup. (c) Let L be the generator of a Feller semigroup in C0 (E) and let {(Ω, F, Px ) , (X(t) : t ≥ 0) , (ϑt : t ≥ 0) , (E, E)} be the corresponding Markov process. For every f ∈ D(L) and for every x ∈ E, the process Z t (1.3) t 7→ f (X(t)) − f (X(0)) − Lf (X(s))ds 0

is a Px -martingale for the filtration (Ft )t≥0 , where each σ-field Ft , t ≥ 0, is (some closureTof) σ (X(u) : u ≤ t). In fact the σ-field Ft may taken to be Ft = s>t σ (X(u) : u ≤ s). It is also possible to

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R complete Ft with respect to Pµ , given by Pµ (A) = Px (A)dµ(x). For Ft the following σ-field may be chosen: \ \ Ft = {Pµ -completion of σ (X(u) : u ≤ s)} . µ∈P (E) s>t

(d) Conversely, let L be a densely defined linear operator with domain D(L) and range R(L) in C0 (E). Let (Px : x ∈ E) be a unique family of probability measures, on an appropriate measure space (Ω, F) with an appropriate filtration (Ft )t≥0 , such that, for all x ∈ E, Px [X(0) = x] = 1, and such that for all f ∈ D(L) the process in (1.3) is a Px -martingale with respect to the filtration (Ft )t≥0 . Then the operator L possesses a unique extension L0 , which generates a Feller semigroup in C0 (E). (e) (Unique Markov extensions) Suppose that the densely defined linear operator L (with domain and range in C0 (E)) possesses the Korovkin property as well as the following one. For every λ > 0 (large) and for every h ∈ D(L), the inequality (1.4)

λ sup h(x) ≤ sup (λI − L) h(x) or, equivalently x∈E

(1.5)

x∈E

λ inf h(x) ≥ inf (λI − L) h(x) x∈E

x∈E

is valid. Then L extends to a unique generator L0 of a Feller semigroup, and the martingale problem is well posed for the operator L. Moreover, the Markov process associated with L0 solves the martingale problem uniquely for L. Definition 3. The operator L possesses the Korovkin property in the sense that there exists a strictly positive real number t0 > 0 such that for every x0 ∈ E ∪ {4} the equality (1.6)

inf

sup {h(x0 ) + [g − (I − t0 L) h] (x)}

h∈D(L) x∈E

(1.7)

= sup inf {h(x0 ) + [g − (I − t0 L) h] (x)} h∈D(L) x∈E

is valid for all g ∈ C0 (E). The proof of assertion (e) is based on the following result. Proposition 4. Let L be a linear operator with range R(L) and 1 domain D(L) in C0 (E). Fix t0 = > 0. Suppose that for every x0 ∈ λ0


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E ∪ {4}, and for every g ∈ C0 (E) the equality sup {h(x0 ) + [g − (I − t0 L) h] (x)}

inf

h∈D(L) x∈E

= sup inf {h(x0 ) + [g − (I − t0 L) h] (x)}

(1.8)

h∈D(L) x∈E

is valid. Also suppose that for every λ > 0 and for every h ∈ D(L) the inequality λ sup h(x) ≤ sup (λI − L) h(x)

(1.9)

x∈E

x∈E

is valid. Then, for 0 < λ < 2λ0 , the following identities are true: λR(λ)g(x0 ) = Λ+ (g, x0 , λ) := inf max h0 ∈D(L),h1 ∈D(L),h2 ∈D(L),... x1 ∈E,x2 ∈E,... (∞ ∞ X λ X λ j λ j hj (xj ) + 1− g(xj+1 ) 1− λ0 λ0 λ0 j=0

j=0

(1.10) −

∞ X j=0

λ 1− λ0

= lim inf

j

) 1 I − L hj (xj+1 ) λ0

inf

max

n→∞ h0 ∈D(L),... ,hn ∈D(L) x1 ∈E,x2 ∈E,... ,xn+1 ∈E

(1.11) "

n λ X λ j−1 λ 1 1− h0 (x0 ) + g(x1 ) − I − L h0 (x1 ) λ0 λ0 λ0 λ0 j=1 n λ X λ j−1 + 1− 1− λ0 λ0 j=1 # λ 1 hj (xj ) + g (xj+1 ) − I − L hj (xj+1 ) λ0 λ0

=

inf h1 ∈D(L),h2 ∈D(L),... ,h0 = λλ

P∞

0

(∞ X j=0

1−

λ λ0

j

hj (xj ) +

j−1

j=1

1− λλ

λ λ0

∞ X

0

j=0

1−

max

hj

x1 ∈E,x2 ∈E,...

λ λ0

j

g(xj+1 )

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(1.12) ∞ X

−

j=0

λ 1− λ0

j

) 1 I − L hj (xj+1 ) λ0

(1.13)

1 = inf max h(x0 ) + g − I − L h (x) λ h∈D(L) x∈E (1.14) 1 = sup min h(x0 ) + g − I − L h (x) λ h∈D(L) x∈E =

sup min j−1 x1 ∈E,x2 ∈E,... ∞ X λ λ 1− hj h1 ∈D(L),h2 ∈D(L),... , h0 = λ0 λ0 j=1 (∞ ∞ X λ j λ X λ j 1− hj (xj ) + 1− g(xj+1 ) λ0 λ0 λ0 j=0

j=0

(1.15) ∞ X

−

j=0

=

λ 1− λ0

j

sup

) 1 I − L hj (xj+1 ) λ0 min

h0 ∈D(L),h1 ∈D(L),h2 ∈D(L),... x1 ∈E,x2 ∈E,...

(∞ X j=0

λ 1− λ0

j

∞ λ X λ j hj (xj ) + 1− g(xj+1 ) λ0 λ0 j=0

(1.16) −

∞ X j=0

= lim inf

λ 1− λ0

j

sup

) 1 I − L hj (xj+1 ) λ0 min

n→∞ h ∈D(L),... ,h ∈D(L) x1 ∈E,x2 ∈E,... ,xn+1 ∈E n 0

(1.17) "

n λ X λ j−1 λ 1 1− h0 (x0 ) + g(x1 ) − I − L h0 (x1 ) λ0 λ0 λ0 λ0 j=1


233

n λ X λ j−1 + 1− 1− λ0 λ0 j=1 # 1 λ hj (xj ) + g (xj+1 ) − I − L hj (xj+1 ) λ0 λ0 ∞ λ j λ X 1− (λ0 R (λ0 ))j+1 g(x0 ). = λ0 λ0

(1.18)

j=0

A discussion of the proof of Theorem 2 can be found in [48]. For a concise formulation of our results we need another definition. 4 4 Definition 5. The operator L with domain D L in C E 4 = C E 4 , R given by D L4 = h ∈ C E 4 : h − h(4) ∈ D(L) , and L4 h = L(h − h(4)), h ∈ D L4 . Here we wrote h(4) = limx→4 h(x). 4 It is noticed that L satisfies the maximum principle in the sense that 4 if h ∈ D L satisfies supx∈E h(x) > h(4), then there exists x0 ∈ E for which h(x0 ) = supx∈E h(x), and for which L4 h(x0 ) ≤ 0. Fix x0 ∈ E and λ > 0. The functional g 7→ Λ+ (g, x0 , λ), g ∈ C0 (E), is defined as follows: Λ+ (g, x0 , λ) 1 4 4 = inf h(x0 ) : h ∈ D L , I − L h≥g λ 1 = inf sup h(x0 ) + g − I − L h (x) λ h∈D(L) x∈E 1 = inf sup min max h(x0 ) + g − I − L h (x) . λ Γ⊂D(L) Φ⊂E h∈Γ x∈Φ∪{4} #Γ 0}, then sup inf {h(x0 ) + [g − (I − t0 L) h] (x)} ≤ λ0 R1 (λ0 )g(x0 )

h∈D(L) x∈E

≤

inf

sup {h(x0 ) + [g − (I − t0 L) h] (x)}

h∈D(L) x∈E

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Jan A. Van Casteren

≤ sup inf {h(x0 ) + [g − (I − t0 L) h] (x)} . h∈D(L) x∈E

The same is true for λ0 R2 (λ0 )g(x0 ). Consequently R1 (λ0 ) = R2 (λ0 ), and thus L1 = L2 . First we give an alternative description of the operator L0 . Again the equality in (1.8) is available. The quantities in (1.13) and (1.14) of Proposition 4 are equal. As a consequence, we may repeat the construction in Proposition 4 for any 0 < λ1 < 2λ0 instead of λ0 . In this way we obtain a resolvent family {R(λ) : 0 < λ < 4λ0 }, for which Proposition 4 is applicable. By induction we find a resolvent family {R(λ) : λ > 0} with the property 1 λR(λ)g(x0 )= inf sup min max h(x0 )+ g − I − L h (x) λ Γ⊂D(L) Φ⊂E h∈Γ x∈Φ∪{4} #Γ 0, and hence g = 0. This proves that L0 is well-defined. Since α (R(β) − R(α)) − R(β) α (αR(α) − I) R(β) = α α−β β α (1.22) =α R(β) − R(α) → βR(β) − I, α−β α−β


239

if α tends to ∞, it follows that L1 R(β) = βR(β) − I. As a consequence we see that L1 extends L0 . Next suppose that g1 belongs to D(L1 ). Then (1.23) (λI − L1 ) g1 =(λI − L0 ) R(λ) (λI − L1 ) g1 =(λI − L1 ) R(λ) (λI − L1 ) g1 . Since, for g ∈ D(L1 ), R(β) (λI − L1 ) g = λR(β)g − lim αR(β) (αR(α) − I) g α→∞

α (αR(α)g − βR(β)) g α−β = λR(β)g + g − βR(β)g = (λI − L0 ) R(β)g. = λR(β)g + lim

α→∞

(1.24)

From (1.23) and (1.24) we obtain 0 = βR(β) (λI − L1 ) (g1 − R(λ) (λI − L1 ) g1 ) (1.25)

= β (λI − L0 ) R(β) (g1 − R(λ) (λI − L1 ) g1 ) .

In (1.25) we let β tend to ∞. Since L0 is a closed linear operator, we obtain that the function g1 − R(λ) (λI − L1 ) g1 belongs to the domain of L0 , and consequently g1 is a member of D(L0 ). Next we show that the operator L0 verifies the maximum principle. Fix g ∈ D(L0 ), and let x0 ∈ E be such that g(x0 ) = supx∈E g(x). Then

(1.26)

L0 g(x0 ) = L1 g(x0 ) = lim α (αR(α)g(x0 ) − g(x0 )) α→∞ ≤ lim sup g(x) − g(x0 ) ≤ 0. α→∞

x∈E

In addition, we show that L0 extends L. If h0 belongs to D(L), then L0 Λ ((I − t0 L)h0 , ·) (x0 ) = λ0 Λ ((I − t0 L)h0 , x0 ) − λ0 (I − t0 L) h0 (x0 ) = λ0 h0 (x0 ) − λ0 h0 (x0 ) + Lh0 (x0 ) = Lh0 (x0 ). Since D(L) is dense, it follows that the domain of L0 is dense as well. Since (λI − L0 ) R(λ) = I we see that the range of λI − L0 coincides with C0 (E). From the Lumer-Phillips theorem, we may conclude that the operator L0 generates a Feller semigroup: see e.g. [44]. Next we prove the uniqueness. Let L1 and L2 be two linear extensions of L which generate Feller semigroups with respective resolvent families {R1 (λ) : λ > 0} and {R2 (λ) : λ > 0}. Then there exists a probability measures µ1x0 and µ2x0 on the Borel field of E 4 such that λ0 Rj (λ0 )g(x0 ) =

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g(y)dµjx0 , g ∈ C0 (E), j = 1, 2. Fix ε > 0. Then we obtain, for some finite subset Γ = Γε of D(L), ε (1.27) − + Λ− (g, x0 ) 2 ε ≤ inf max min {h(x0 ) + [g − (I − t0 L) h] (x)} − h∈Γ 4 x∈Φ∪{4} Φ⊂E R

#Φ 0 is arbitrary we conclude R1 (λ0 )g(x0 ) = R2 (λ0 )g(x0 ), g ∈ C0 (E), x0 ∈ E, and hence R1 (λ0 ) = R2 (λ0 ). Thus L1 = L2 . Let L0 be the (unique) extension of L, which generates a Feller semigroup, and let {(Ω, F, Px ) , (X(t), t ≥ 0) , (ϑt , t ≥ 0) , (E, E)} be the corresponding Markov process with Ex [g(X(t)] = exp (tL0 ) g(x), g ∈ C0 (E), x ∈ E, t ≥ 0. Then the family {Px : x ∈ E} is a solution to the martingale problem associated to L. The proof of the uniqueness part follows a pattern similar o to the n proof of the uniqueness part (e) of n o (1) (2) Theorem 2. Let Px : x ∈ E and Px : x ∈ E be two solutions to the martingale problem for L. Fix x0 ∈ E, g ∈ C0 (E), and s > 0. Then, as in the proof of the first part of (e) of Theorem 2 Z ∞ Λ− (g, X(s), λ) ≤ λ exp(−λt)E(j) x0 g (X(t + s)) Fs dt 0

≤ Λ+ (g, X(s), λ) ,

for j = 1, 2, where +

Λ (g, x0 , λ)= inf

1 sup min max h(x0 )+ g − I − L h (x) ; λ Φ⊂E h∈Γ x∈Φ∪{4}

Γ⊂D(L) #Γ 0 : X(s) = 4} . Then ζ is called the life time of the process {X(t) : t ≥ 0}. The motion {X(t) : t ≥ 0} is Px -almost surely right continuous and possesses left limits in E on its life time: (i) lims↓t X(s) = X(t), (right continuity); (ii) s ≥ t, X(t) = 4, implies X(s) = 4, (4 is called the cemetery); (iii) lims↑t X(s) = X(t−) ∈ E, t < ζ, (left limits in E on its life time). These assertions hold Px -almost surely for all x ∈ E. The probability P4 may be defined by P4 (A) = δω4 (A), where ω4 (s) = 4, s > 0. Remark 4. The shift or translation operators ϑs : Ω → Ω, s ≥ 0, possess the property that X(t) ◦ ϑs = X(t + s), Px -almost surely, for all x ∈ E and for all s and t ≥ 0. This is an extremely important property. For example f (X(t)) ◦ ϑs = f (X(t + s)), f ∈ C0 (E), s, t ≥ 0. If Ω is the Skorohod space Ω = D [0, ∞], E 4 , then X(t)(ω) = ω(t) = X(t, ω) = ω(t), ϑt (ω)(s) = ω(s + t), ω ∈ Ω. Remark 5. For every x ∈ E, the measure Px is a probability measure on F with the property that Px [X(0) = x] = 1. So the process starts at X(0) = x, Px -almost surely, at t = 0. This is the normality property. Remark 6. The Markov property can be expressed as follows: Ex f (X(s + t)) Fs (1.31) =Ex f (X(s + t)) σ(X(s)) =EX(s) [f (X(t))] ,

Px -almost surelyfor all f ∈ C0 (E) and for all s and t ≥ 0. Of course, the expression E Y F denotes conditional expectation. The meaning of Ft is explained in Remark 2. Let Y : Ω → C be a bounded random variable. This means that Y is measurable with respect to the field generated by {X(u) : u ≥ 0}. The Markov property is then equivalent to (1.32) Ex Y ◦ ϑs Fs = EX(s) [Y ] , Px -almost surely for all bounded random variables Y and for all s ≥ 0. Notice that, intuitively speaking, Fs is the information from the past,


243

σ (X(s)) is the information at the present, and Y ◦ ϑs is measurable with respect to some completion of σ {X(u) : u ≥ s}, the information from the future. Put P (t, x, B) = Px [X(t) ∈ B]. Then Ex [f (X(t))] = R f (y)P (t, x, dy), f ∈ C0 (E). Moreover (1.31) is equivalent to (1.32) and to   n Y fj (X(tj )) (1.33) Ex  j=1

=

Z Z

...

Z Y n

(fj (xj )P (tj − tj−1 , xj−1 , dxj )) ,

j=1

for all 0 = t0 ≤ t1 < t2 < · · · < tn < ∞ and for all f1 , . . . , fn in C0 (E). Remark 7. Since the paths {X(t) : t ≥ 0} are right continuous Px almost surely, it can be proved that our Markov process is in fact a strong Markov process. Let S : Ω → ∞ be a stopping time meaning that for every t ≥ 0 the event {S ≤ t} belongs to Ft . This is the same as saying that the process t 7→ 1[S≤t] is adapted. Let FS be the natural σ-field associated with the stopping time S, i.e. \ FS = {A ∈ F : A ∩ {S ≤ t} ∈ Ft } . t≥0

Define ϑS (ω) by ϑS (ω) = ϑS(ω) (ω). Consider FS as the information from the past, σ X(S) as information from the present, and σ {X(t) ◦ ϑS : t ≥ 0} = σ {X(t + S) : t ≥ 0} as the information from the future. The strong Markov property can be expressed as follows: (1.34)

Ex [Y ◦ ϑS |FS ] =EX(S) [Y ] , Px -almost surely on the event {S < ∞},

for all bounded random variables Y , for all stopping times S, and for all x ∈ E. One can prove that under the ”cadlag” property events like {X(S) ∈ B, S < ∞}, B Borel, are FS -measurable. The passage from (1.34) to (1.31) is easy: put Y = f (X(t)) and S(ω) = s, ω ∈ Ω. The other way around is much more intricate and uses the cadlag property of the process {X(t) : t ≥ 0}. In this procedure the stopping time S is approximated by a decreasing sequence of discrete stopping times (Sn = 2−n d2n Se : n ∈ N). The equality Ex [Y ◦ ϑSn |FSn ] = EX(Sn ) [Y ] , Px -almost surely,

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Jan A. Van Casteren

is a consequence of (1.31) for a fixed time. Let n tend to infinity in (1) to obtain (1.34). The “strong Markov property” can be extended to the “strong time dependent Markov property”: ω 0 7→ Y S(ω)+T ω 0 , ω 0 , Ex Y (S+T ◦ ϑS , ϑS ) FS (ω)=E X S(ω)

Px -almost surely on the event {S < ∞}. Here Y : [0, ∞) × Ω → C is a bounded random variable. The Cartesian product [0, ∞) × Ω is supplied with the product field B⊗F; B is the Borel field of [0, ∞) and F is (some extension of) σ (X(u) : u ≥ 0). Important stopping times are “hitting times”, or times related to hitting times: n o T = inf s > 0 : X(s) ∈ E 4 \ Σ , and

S = inf s > 0 :

Z

s

1E\Σ (X(u))du > 0 ,

0

where Σ is some open (or Borel) subset of E 4 . This kind of stopping times have the extra advantage of being terminal stopping times, i.e. t+S◦ϑt = S Px -almost surely on the event {S > t}. A similar statement holds for the hitting time T . The time S is called the penetration time of E \ Σ. Let p : E → [0, ∞) be a Borel measurable function. Stopping times of the form Z s Sξ = inf s > 0 : p X(u) du > ξ 0

serve as a stochastic time change, because they enjoy the equality: Sξ + Sη ◦ ϑSξ = Sξ+η , Px -almost surely on the event {Sξ < ∞}. As a consequence operators of the form S(ξ)f (x) := Ex [f (X (Sξ ))], f a bounded Borel function, possess the semigroup property. Also notice that S0 = 0, provided that the function p is strictly positive. Remark 8. Since a Feller semigroup possesses a generator, L say, one also says that L generates the associated strong Markov process. For example 21 ∆ generates Brownian motion. This concept yields a direct relation between certain (lower order) pseudo-differential operators and probability theory: see Jacob [26]. The order has to be less than or equal to 2. This follows from the theory of Lévy processes and the Lévy-Khinchin formula, which decomposes a continuous negativedefinite function into a linear term (probabilistically this corresponds to a deterministic drift), a quadratic term (this corresponds to a diffusion: a continuous Brownian motion-like process), and a term that corresponds


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to the jumps of the process (compound Poisson process, Lévy measure). Quite a number of problems in classical analysis can be reformulated in probabilistic terms. For more work on the connection between the martingale problem and (pseudo-)differential operators, the reader may consult e.g. papers by Hoh [23, 24], and papers by Mikulyavichyus and Pragarauskas [33, 34]. For a connection between the martingale problem and quadratic forms, see e.g. Albeverio and Röckner [3]. For a relationship between the maximum principle and Dirichlet operators see Schilling [38]. For instance for certain Dirichlet boundary value problems hitting times are appropriate; for certain initial value problems Markov process theory is relevant. For other problems the martingale approach is more to the point. For example there exists a one-to-one correspondence between the following concepts: (i) Unique (weak) solutions of stochastic differential equations in Rν : (ii) Unique solutions to the corresponding martingale problem; (iii) Markovian diffusion semigroups in Rν ; (iv) Feller semigroups generated by certain second order differential operators of elliptic type. For more details see e.g. Ikeda and Watanabe [25]. (Regular) first order perturbations of second order elliptic differential operators can be studied using the Cameron-Martin-Girsanov transformation. Perturbations of order zero are treated via the Feynman-Kac formula. Remark 9. In our discussion we started with (generators of) Feller semigroups. Another approach would be to begin with symmetric Dirichlet forms (quadratic form theory) in L2 (E, m), where m is a Radon measure on the Borel field E of E. (By definition a Radon measure assigns finite values to compact subsets and it is inner and outer regular.) The reader may consult the books by Bouleau and Hirsch [6], by Fukushima, Oshima and Takeda, [21], or by Z. Ma and M. Röckner [31]. In the latter reference Ma and R¨ ockner treat somewhat more general Dirichlet forms. These Dirichlet need not be symmetric, but they obey a certain cone type inequality: |E(f, g)|2 ≤ KE(f, f )E(g, g),

f, g ∈ D (E) .

Again one says that the Markov process is generated by (or associated to the Dirichlet form E or to the corresponding closed linear operator: E(f, g) = − hLf, gi, f ∈ D(L), g ∈ D (E). (Note that only regular Dirichlet forms correspond to Markov processes.)

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Problem 1. (a) Is a result like Theorem 2 true if the locally compact space E is replaced with a Polish space, and if Cb (E) (space of all bounded continuous functions on E) replaces C0 (E)? Instead of the topology of uniform convergence we consider the strict topology. This topology is generated by seminorms of the form: f 7→ supx∈E |u(x)f (x)|, f ∈ Cb (E). The functions u ≥ 0 have the property that for every α > 0 the set {u ≥ α} is compact (or is contained in a compact) subset of E. The functions u need not be continuous. What about Markov uniqueness? Is there a relationship with work done by Eberle [17, 18, 19]? (b) Is it possible to rephrase Theorem 2 for reciprocal Markov processes and diffusions? Martingales should then replaced with differences of forward and backward martingales. A stochastic process (M (t) : t ≥ 0) on a probability space (Ω, F, P) is called a backward martingale if E[M (t) Fs ] = M (s), P-almost surely, where t < s, and Fs is the σ-field generated by the information from the future: Fs = σ (X(u) : u ≥ s}. Of course we assume that M (t) belongs to L1 (Ω, F, P), t ≥ 0. Let (Ω, F, P) be a probability space. An E-valued process (X(t) : 0 ≤ t ≤ 1) is called a reciprocal Markov process if for any 0 ≤ s < t ≤ 1 and every pair of events A ∈ σ (X(τ ) : τ ∈ (s, t)), B ∈ σ (X(τ ) : τ ∈ [0, s] ∪ [t, 1]) the equality (1.35) P A ∩ B X(s), X(t) = P A X(s), X(t) P B X(s), X(t) is valid. By D we denote the set (1.36) D = {(s, x, t, B, u, z) : (x, z) ∈ E × E, 0 ≤ s < t < u ≤ 1, B ∈ E} . A function P : D → [0, ∞) is called a reciprocal probability distribution or a Bernstein probability if the following conditions are satisfied: (i) the mapping B 7→ P (s, x, t, B, u, z) is a probability measure on E for any (x, z) ∈ E × E and for any 0 ≤ s < t < u ≤ 1; (ii) the function (x, z) 7→ P (s, x, t, B, u, z) is E ⊗ Ecal-measurable for any 0 ≤ s < t < u ≤ 1;


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(iii) For every pair (C, D) ∈ E ⊗ E, (x, y) ∈ E × E, and for all 0 ≤ s < t < u ≤ 1 the following equality is valid: Z P (s, x, u, dξ, v, y) P (s, x, t, C, u, ξ) D Z = P (s, x, t, dη, v, y) P (t, η, u, C, v, y) . C

Then the following theorem is valid for E = Rν (see Jamison [27]). Theorem 7. Let P (s, x, t, B, u, y) be a reciprocal transition probability function and let µ be a probability measure on E ⊗ E. Then there exists a unique probability measure Pµ on F with the following properties: (1) With respect to Pµ the process (X(t) : 0 ≤ t ≤ 1) is reciprocal; (2) For all (A, B) ∈ E ⊗ E the equality Pµ[X0 ∈A, X1 ∈B]=µ (A×B) is valid; (3) For every 0 ≤ s < t < u ≤ 1 and for every A ∈ E the equality Pµ X(t) ∈ A X(s), X(u) = P (s, X(s), t, A, u, X(u)) is valid. For more details see Thieullen [39] and [40]. (c) What is the equivalent of all this in the non-commutative setting? We notice that there is a possibility to define a strict topology on a C ∗ algebra. To be precise, we let A be a C ∗ -algebra, and π : A → L (H) be a faithful representation (obtained e.g. via a Gelfand-Naimark-Segal construction). Define, for every compact operator T : H → H the seminorm pT : A → [0, ∞) by pT (x) = kT π(x)k, x ∈ A. The topology τβ induced by these seminorms may be called the strict topology. We notice that a positive solution to Problem 1 (a) would make a nice link with work by Dorroh and Neuberger [16] in as much as the Lie generator will also be the generator of a Markov process. Problem 2. How useful is the martingale result on operators with the Korovkin property? We don’t have a good example or application. Is there a non-commutative version of the Korovkin property?

2. Feynman-Kac semigroups: 0-order regular perturbations In the present section we will quote one central theorem from [15]. We assume that V : E → [−∞, ∞] is a Kato-Feller potential function in

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the following sense: (2.1)

lim sup t↓0 x∈E

lim sup t↓0 x∈E

Z

t

Ex V− X(τ )

0

Z 0

dτ = 0;

t

Ex 1K X(τ ) V+ X(τ ) dτ = 0

for all compact subsets K of E, We note that a Kato-Feller potential function belongs to L1loc (E, m). Furthermore we consider Σ: a (large) open subset of E, with complement Γ. The symbol S denotes the penetration time of Γ: Z s (2.2) S = inf s > 0 : 1Γ (X(u))du > 0 0

It is a blanket assumption that p0 (t, x, y) = p0 (t, y, x) (symmetry), and that the function (t, x, y) 7→ p0 (t, x, y) is continuous on (0, ∞) × E × E. In addition dx = dm(x) is a reference measure on E. The operator −K0 is supposed to generate a symmetric strong Markov process {(Ω, F, Px ) , (X(t) : t ≥ 0) , (ϑt : t ≥ 0) , (E, E)} . Z So the law Px (X(t) ∈ B) is given by Px (X(t)∈B)= p0 (t, x, y)dm(y). B

Definition 8. The pinned measure (2.3)

µt,y 0,x

on Ft− is defined as follows:

µt,y 0,x (A) = Ex [p0 (t − s, X(s), y)1A ] ,

where A ∈ Fs , s < t. Extend this pre-measure to a genuine measure on Ft− and notice that the process s 7→ p0 (t − s, X(s), y) is a martingale. The measure µt,y 0,x lives on the event {X(0) = x, X(t−) = y}. Indeed, it follows from the Kolmogorov extension theorem on cylindrical measures that the measure µt,y 0,x , determined by (2.3), can be extended to the σ-field Ft− . Since the process s 7→ p0 (t − s, X(s), y) is a Px -martingale on the interval 0 ≤ s < t, it follows that the quantity µt,y 0,x (A) is well-defined: its value does not depend on s, as long as A belongs to Fs and s < t. Theorem 9. ˙ extend(a) There exist a closed densely defined linear operator K0 +V ing K0 + V , which generates a positivity preserving (self-adjoint) semigroup in L2 (E, m), denoted by ˙ exp −t K0 +V :t≥0 .


249

This semigroup is given by the Feynman-Kac formula (f ∈ L2 (E, m)): (2.4) Z t ˙ exp −t K0 +V f (x) = Ex exp − V (X(u))du f (X(t)) . 0

˙ (b) Every operator exp −t K0 +V is an integral operator with a continuous, symmetric integral kernel exp(−t(K0+V ))(x, y) given by

(2.5) (2.6)

exp (−t (K0 + V )) (x, y) Z s = lim Ex exp − V (X(u))du p0 (t − s, X(s), y) s↑t 0 Z t Z = exp − V (X(u))du dµt,y 0,x . 0

(c) The quadratic form (generalized Schr¨ odinger form) EV associated with the above Feynman-Kac semigroup is given by (2.7) EV (f, g) =

Dp E Dp E Dp E p p p K0 f, K0 g + V+ f, V+ g − V− f, V− g ,

for f , g members of Z p 2 2 (2.8) D K0 ∩ f ∈ L (E, m), V+ (x) |f (x)| dm(x) < ∞ . For a proof we refer the interested reader to Chapter 2 in [15]. 3. Dirichlet semigroups: 0-order singular perturbations, harmonic functions In this section we present some results concerning Feynman-Kac semigroups, but with infintely high potentials in certain regions (obstacles, potential barriers): the repulsive part of the potentials takes the value ∞ in such regions. ˙ :t≥0 Definition 10. The Dirichlet semigroup exp − K0 +V Σ is defined by (3.1) ˙ exp −t K0 +V

Σ

Z t i f (x) = Ex exp − V (X(u))du f (X(t)) : S > t . h

0

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The V -harmonic extension operator HΣV is defined by Z S V (3.2) HΣ f (x) = Ex exp − V (X(u))du f (X(S)) : S < ∞ . 0

We repeat that as in equality (2.2) the symbol S denotes the peneR∞ tration time of the set Γ := E \ Σ. We have HΣa+V = a 0 e−as TΣV (s)ds, where Z S V (3.3) TΣ (s)f (x) = Ex exp − V (X(u))du f (X(S)) : S < s . 0

Theorem 11. The operator −1/2 1/2 a+V ˙ ˙ (3.4) HΣ aI + K0 +V aI + K0 +V extends to an orthogonal projection and Dynkin’s formula (3.5) ˙ aI + K0 +V

−1

˙ = J ∗ aI + K0 +V

−1 Σ

˙ J + HΣa+V aI + K0 +V

−1

is valid. At the level of integral kernels, the latter formula reads: Newton potential = Green function + Harmonic correction. We also have a+V a+V ˙ D Ea+V = D aI + K0 +V (3.6) ⊕ : a+V R H Σ Σ HΣa+V is an orthogonal projection in D Ea+V , endowed with the canonical quadratic form Ea+V corresponding to the Feynman-Kac semigroup corresponds, and which is given by D E D E 1/2 1/2 1/2 1/2 Ea+V (f, g) = K0 f, K0 g + V+ f, V+ g D E 1/2 1/2 + a hf, gi − V− f, V− g , 1/2

1/2

where f and g belong to D(K0 )∩D(V+ ). A proof of Dynkin’s formula can be found in Chapter 2 of [35], and the assertion about the projection HΣa+V is proved in detail in [46]. Put Z S a+V a+V (a + V (X(u))) du : S < ∞ . hΣ (x) = HΣ 1(x) = Ex exp − 0


251

R Definition 12. The set E\Σ has finite a+V -capacity if ha+V (x)dx Σ is finite. Note: Z (3.7) ha+V (x)dx = inf Ea+V (u, u) : u ∈ D Ea+V , u ≥ 1Γ . Σ The following theorem appears in [47] as Theorem 3.1, and in [46] as Theorem 13 on page 176. Theorem 13. Let {E0 (ξ) : ξ ∈ R} and {E1 (ξ) : ξ ∈ R} be the spec˙ and to K0 +W ˙ , respectively. tral decompositions corresponding to K0 +V Let {(Ω, F, Px ) , (X(t) : t ≥ 0) , (ϑt : t ≥ 0) , (E, E)} be the strong Markov process generated by −K0 . Suppose that, for some t0 > 0, the function exp (−t0 K0 ) |W − V | is bounded, or suppose that "Z 2 # t (3.8) lim sup Ex (W (X(u)) − V (x(u))) du = 0. t↓0 x∈E

0

The following assertions are equivalent: (i) For every bounded interval A the operator E0 (A)(W − V )E1 (A) is compact; (ii) For some t > 0 (for all t > 0) the operator ˙ ˙ exp −t K0 +V (W − V ) exp −t K0 +W is compact; (iii) For some t > 0 (for all t > 0), the operator D(t) is compact. Rt Remark 1. If limt↓0 supx∈E 0 [exp (−sK0 ) |W − V |] (x)ds = 0, then "Z 2 # t (3.9) lim sup Ex (W (X(u)) − V (x(u))) du = 0. t↓0 x∈E

0

This is a consequence of the Markov property. Remark 2. An equality like (3.9) can probably be used for first order perturbations, where the Cameron-Martin formula is applicable. In such a case Z we have to deal with stochastic integrals instead of the t

(W (X(u)) − V (X(u))) du.

process t 7→

0

Remark 3. Theorem 13 is probably not known, even in the case where we consider K0 = H0 = − 21 ∆. The corresponding process is Brownian motion in this case.

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Remark 4. We introduced Brownian motion as a Markov process with a certain transition function. It can also be introduced as a νdimensional Gaussian process {X(t) : t ≥ 0} such that E [hX(t), X(s)i] = ν min(s, t), or as a Lèvy process with negative definite function ξ 7→ 2 1 2 |ξ| , or as a martingale with variation process t 7→ t. Remark 5. In the implication (iii) ⇒ (ii) the following identity is relevant: t t ˙ ˙ (3.10) t exp − K0 +V (W − V ) exp − K0 +W 2 2 Z π ∞ 1 ˙ = exp iτ t K +V D(t)(W − V ) 0 2 −∞ (cosh (πτ ))2 ˙ exp −iτ t K0 +W dτ, where (3.11)

D(t)T =

Z

t

˙ exp −u K0 +V

0

˙ T exp −(t − u) K0 +W

du.

The following result is applicable for Z t M (t) = exp − V (X(u))du 0

or Z t M (t) = exp − V (X(u))du 1{S>t} , 0

where V is a Kato-Feller potential, and where S is a terminal stopping time, i.e. t+S ◦ϑt = S Px -almost surely on the event {S > t}. Theorem 14 proves part of Theorem 9. The proof of the following result can be found in [46], Theorem 14, page 181. Theorem 14. Let {M (t) : t ≥ 0} be a multiplicative process taking its values in [0, ∞). This means that, for every t ≥ 0, M (t) : Ω → [0, ∞) is Ft -measurable and that M (s + t) = M (s)M (t) ◦ ϑs for all s and t ≥ 0. Assume Z Z lim M (t − )dµt,y = M (t)dµt,y 0,x 0,x . ↓0

As above, the defining property of µt,y 0,x is the equality Z F dµt,y 0,x = Ex [F p0 (t − s, X(s), y)] ,


253

where F : Ω → R is bounded and Fs -measurable (s < t). The following assertions are valid: (1) The process Z s 7→ M (s) M (t − s)dµt−s,y 0,X(s) is a Px -martingale on the interval [0, t). (2) The following equality is valid: Z Z (3.12) Ex [M (t)f (X(t))] = M (t)dµt,y 0,x f (y)dy, where f is greater than or equal to zero and Borel measurable. (3) The following Chapman-Kolmogorov identity is valid: Z Z Z Z t1 ,z t2 ,y 1 +t2 ,y (3.13) M (t1 )dµ0,x M (t2 )dµ0,z dz = M (t1 + t2 ) dµt0,x . Problem 3. Let M (t) be as in Theorem 14. Define the semigroup exp (−tKM ) by Z [exp (−tKM ) f ] (x) = exp (−tKM ) (x, y)f (y)dy, R where exp (−tKM ) (x, y) = M (t)dµt,y 0,x . Suppose that the operators exp (−tKM ), t > 0, are self-adjoint. Then, formally, Z [exp (−isKM ) exp (−tKM ) (·, y)] (x)f (y)dy Z = exp (−isKM ) exp (−tKM ) (·, y)f (y)dy (x) = [exp (−isKM ) exp (−tKM ) f ] (x) = [exp (−(t + is)KM ) f ] (x). In what sense do we have convergence of [exp (−isKM)exp (−tKM) (·, y)] (x) to exp (−isKM) (x, y), if t tends to 0 downward? Remark 1. In relation to the previous problem, we like to point out that Zambrini and coworkers [52, 2, 41, 42] have kind of a transition scheme to go from classical stochastic calculus (with non-reversible processes) to physical real time (reversibile) quantum mechanics and vice versa. An important tool in this connection is the so-called Noether theorem. In fact, in Zambrini’s words, reference [52] contains the first concrete application of this theorem. In [52] the author formulates a theorem like Theorem 15 below, he also uses so-called ”Bernstein diffusions” (see e.g. [10]) for the ”Euclidean Born interpretation” of quantum mechanics. The Bernstein diffusions are related to solutions of

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Jan A. Van Casteren

∂ ∂ ˙ ˙ − K0 +V η = 0, and of + K0 +V η ∗ = 0. It would be nice ∂t ∂t to formulate the stochastic Noether theorem (Theorem 2.4 in [52]) in terms of the carré du champ operator and ideas from stochastic control.

In relation to the previous remark we have the following result for generators of diffusions: see Remark 2 following Theorem 27. For the notion of the squared gradient operator (carré du champ operateur) see equality (5.10). The operator K0 acts on the first variable and so does the squared gradient operator Γ1 . Theorem 15. Let χ : E × [t, u] → [0, ∞] be a function such that [|log χ (X(u), u)|] , v ∈ D (K0 ) is finite. Here u > 0 is a fixed time and {(Ω, F, Px ) , (X(t) : t ≥ 0) , (ϑt : t ≥ 0) , (E, E)} is the strong Markov process generated by −K0 . Let SL be a solution to the following Riccati type equation. (This equation is called the Hamilton-JacobiBellman equation.) For t ≤ s ≤ u and x ∈ E the following identity is true: (3.14)  1  ∂SL (x, s) + Γ1 (SL , SL ) (x, s) + K0 SL (x, s) − V (x, s) = 0; − ∂s 2 S (x, u) = − log χ(x, u), x ∈ E. L M Ex,tv,t

Then for any real valued v ∈ D (K0 ) the following inequality is valid: (3.15) M SL (x, t)≤Ex,tv,t

Z u t

1 M Γ1 (v, v)+V (X(τ ), τ )dτ −Ex,tv,t [χ (X(u), u)] , 2

and equality is attained for the “Lagrangian action” v = SL . By definition Ex,t [Y ] is the expectation, conditoned at X(t) = x, of the random variable Y which is measurable with respect to the information from the future: i.e. with respect to σ {X(s) : s ≥ t}. The measure M Px,tv,t is defined in equality (3.17) below. Put ηχ = exp (−SL ), where ∂ ˙ SL satisfies (3.14). From (5.21) it follows that − K0 +V ηχ = 0, R ∂t provided that K0 1 is interpreted as 0, i.e. K0 f dm = 0 for all f ∈ ∂ D (K0 ). Fix a function v : E × R → R in D (K0 − D1 ), where D1 = ∂t is differentiation with respect to t. Let the process {(Ω, F, Px,t ) , ((qv (t), t) : t ≥ 0) , (ϑt : t ≥ 0) , (E × R, E ⊗ B)}


255

be the Markov process generated by the operator −Kv + D1 , where Kv is defined by Kv (f )(x, t) = K0 f (x, t) + Γ1 (v, f ) (x, t). Here, B denotes the Borel field of R, and by Γ1 (v, f )(x, t) we mean (3.16) 1 Γ1 (v, f )(x, t)=lim Ex [(v(X(s), t)−v(X(0), t)) (f (X(s), t)−f (X(0), t))] . s↓0 s We also believe that the following version of the Cameron-Martin formula is valid. For all finite n-tuples t1 , . . . , tn in (0, ∞) the identity (3.18) is valid:   n Y M Ex,tv,t  fj (X (tj + t) , tj + t) (3.17) j=1

=Ex,t

h

Z 1 u exp − Γ1 (v, v) (X(τ ), τ ) dτ − Mv,t (u) 2 t n i Y fj (X (tj + t) , tj + t)

j=1

(3.18)



=Ex,t 

n Y

j=1



fj (qv (tj + t) , tj + t)

where the Ex,t -martingale Mv,t (s), s ≥ t, is given by (3.19) Mv,t (s) = v (X(s), s) − v (X(t), t) +

Z t

s

∂ − + K0 v (X(τ ), τ ) dτ. ∂τ

Its quadratic variation part hMv,t i (s) := hMv,t , Mv,t i (s) is given by Z s (3.20) hMv,t i (s) = Γ1 (v, v) (X(τ ), τ ) dτ. t M

The equality in (3.17) serves as a definition of the measure Px,tv,t (·), and the equality in (3.18) is a statement. We notice that the following processes are Px,t martingales on the interval [t, u]: 1 (3.21) exp − hMv,t i (s) − Mv,t (s) and 2 1 (3.22) exp − hMv,t i (s) − Mv,t (s) (hMv,t i (s) + Mv,t (s)) . 2

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Proof. The proof is based on the following version of Jensen’s inquality and should be compared with the arguments in Zambrini [52], who used ideas from Fleming and Soner: see Chapter VI in [20]. The inequality we have in mind is the following one: M

M

− log Ex,tv,t [exp (−ϕ)] ≤ Ex,tv,t [ϕ] ,

(3.23)

with equality only if ϕ is constant Px,t -almost surely. We apply (3.23) for the stochastic variable ϕ = ϕv , given by (3.24) ϕv = −

Z t

u

1 Γ1 (v, v) + V (X(τ ), τ ) dτ − Mv,t (u) − log χ (X(u)) . 2

By Jensen’s inequality we have Z u Mv,t 1 Ex,t hMv,t i (u) + V (X(τ ), τ ) dτ − log χ (X(u), u) 2 t M

(the process in (3.22) is a = Px,tv,t -martingale) (3.25) M Ex,tv,t

Z u 1 − hMv,t i (u) − Mv,t (u) + V (X(τ ), τ ) dτ − log χ (X(u), u) 2 t

(here we apply Jensen’s inequality) h 1 M ≥ − log Ex,tv,t exp hMv,t i (u) + Mv,t (u) 2 Z u i − V (X(τ ), τ ) dτ + log χ (X(u), u) t

M

(definition of the probability measure Ex,tv,t ) Z u = − log Ex,t exp − V (X(τ ), τ ) dτ + log χ (X(u), u) t

(the function SL (y, s) obeys the Hamilton-Jacobi-Bellmann equation (3.14)) Z h 1 u = − log Ex,t exp − Γ1 (SL , SL ) (X(τ ), τ ) dτ − MSL ,t (u) 2 t i × exp (−SL (X(t), t) + SL (X(u), u) + log χ (X(u), u))


(the quadratic variation process hMSL ,t i (s) is given by τ ), τ )dτ ) h 1 = − log Ex,t exp − hMSL ,t i (u) − MSL ,t (u) 2 (3.26)

Rs t

257

Γ1 (SL , SL )(X(

i × exp (−SL (X(t), t) + SL (X(u), u) + log χ (X(u), u)) = SL (x, t). Since we have X(t) = x, Px,t -almost surely, since SL (X(u), u) +log χ (X(u), u) = 0, and since the process exp − 21 hMv,t i (u) − Mv,t (u) is a Px,t -martingale, we see that the expression in (3.26) is equal to SL (x, t). This proves the inequality part of the theorem. If v = SL , then the Hamilton-Jacobi-Bellmann equation implies that the expression in (3.25) equals SL (x, t). Here we employ again the identity Z s hMSL ,t i (s) = Γ1 (SL , SL ) (X(τ ), τ ) dτ. t

Altogether this proves Theorem 15. Problem 4. Prove Theorem 15 for viscosity solutions of the equation (3.27) H x, s1 , s2 , v(x, ·), (Γ1 (v, v) (x, ·))1/2 , −K0 v (x, ·) = 0, where the function H (x, s1 , s2 , v(·), p(·), M (·)) is defined by H (x, s1 , s2 , v(·), p(·), M (·)) (3.28) 1 = v(s1 ) − v(s2 ) + 2

Z

s2

s1

2

p(s) ds −

Z

s2

s1

M (s)ds −

Z

s2

V (x, s)ds.

s1

Here a viscosity solution is defined as a function SL for which (3.29) H x0 , s1 , s2 , ϕ (x0 , ·) , (Γ1 (ϕ, ϕ) (x0 , ·))1/2 , −K0 ϕ (x, ·) ≥ 0, whenever ϕ belongs to D (K0 ) and possesses the property that (3.30) SL (x0 , s) − ϕ (x0 , s) ≤ SL (x, s) − ϕ (x, s) , for all s1 ≤ s ≤ s2 , and for all x in a neighborhood of x0 (or for all x ∈ E). Moreover, if the inequality in (3.30) is reversed, then the one in (3.29) should also be reversed.

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Formulate and prove a stochastic Noether theorem in terms of the squared gradient operator: see Zambrini [52]. It should read something like what follows ((x, s) ∈ E × [t, u], h˙ denotes the time derivative of h). Theorem 16. Suppose that the functions h, SL : E × R → C, T : [0, ∞) → [0, ∞) satisfy ˙ ˙ h(x.s) − h(x, t) − K0 h(x, s) + K0 h(x, t) Z s (3.31) = Γ1 (h, V )(x, σ)dσ + V (x, s)T (s) − V (x, t)T (t); t

1 S˙L (x, s) = K0 SL (x, s) + Γ1 (SL , SL ) (x, s) − V (x, s). 2 Then the proccess h i (3.32) Γ1 (SL , h) + S˙L T + h˙ (qSL (s), s) , t ≤ s ≤ u, is a Ex,t -martingale. Under suitable conditions on the function v : E × [t, u] → C the process (3.33) 1 ˙ Γ1 (v, h) + K0 v + Γ1 (v, v) − V T + h (qv (s), s) , t ≤ s ≤ u, 2 is a Ex,t -martingale as well. Remark 2. There is also a connection with work by Albeverio, Johnson and Ma [1], and Lim [30] about Feynman operational calculus for Kato-Feller potentials. In her work Lim extends the Feynman operational calculus to so-called smooth Kato-Feller measures. In [28] the authors, G. W. Johnson and M. L. Lapidus treat the Feynman Calculus in great length. Corollary 17. Let −KM be the generator of the semigroup (exp (−tKM )) defined as in Theorem 14. The following processes are martingales: (3.34) (3.35)

τ 7→ M (τ ) exp (−(t − τ )KM ) (X(τ ), y), 0 ≤ τ < t; Z τ M (τ )f X(τ ) − f (X(0)) + M (v)KM f X(v) dv. 0

Corollary 18. Let S be a terminal stopping time, and put (3.36)

Σ = {x ∈ E : Px [S = 0] = 0} , Γ = E \ Σ;

(3.37) exp (−t (KM )S ) f (x) = Ex [M (t)f (X(t)) : S > t] ;


259

HSM f (x) = Ex [M (S)f (X(S)) : S < ∞] ;

(3.38)

DS (t) = exp (−tKM ) − JS∗ exp (−t (KM )S ) JS ; Z t DS (t)T = exp (−u (KM )S ) T exp (−(t − u)KM ) du.

(3.39) (3.40)

0

Here JS f = f Σ and JS∗ is its adjoint. Then (singular Duhamel’s formula) DS (t) − 1Γ exp (−tKM ) = JS∗ DS (t) (KM )S JS HSM (3.41) = 1Σ HSM exp (−tKM ) − JS∗ exp (−t (KM )S ) JS HSM + JS∗ DS (t)HSM KM . Khas’minskii’s lemma is available for multiplicative processes. Theorem 19. (Khas’minskii’s Lemma) Let W : E hR → [0, ∞] be ia t Borel measurable function. Put γ = limt↓0 supx∈E Ex 0 W (X(s))ds , and suppose γ < 1. The following assertions are true: (1) γ = lima→∞ supx∈E (aI + K0 )−1 W (x). (2) Choose t0 > 0 in such a way that Z t0 α := sup Ex W (X(s))ds < 1. x∈E

0

Then Z sup Ex exp

x∈E

t0

W (X(s))ds

0

≤

1 . 1−α

1 (3) Let t0 and α be as in (2). Put M = and eb = 1−α Then, for x ∈ E and t ≥ 0, Z t Ex exp W (X(s))ds ≤ M exp(bt).

1 1−α

1/t0

.

0

(4) Let {M (t) : t ≥ 0} be a multiplicative functional attaining values in [0, ∞]. Suppose that inf inf η sup Py sup M (s) ≥ η < 1. η>0 t>0

y∈E

0≤s≤t

Then there exist constants M and b such that Ex [M (t)] ≤ M ebt , x ∈ E, t ≥ 0.

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The proof of (4) can be based on stopping times of the form ξ Tξ = inf t > 0 : sup M (s) > e . 0≤s≤t

Then Tξ + Tη ◦ ϑTξ = Tξ+η . 4. Sets of finite capacity, wave operators, and related results In the present section we collect without proof some of the results obtained in [15], where the proofs can be found as well. The operator J = JΣ denotes the restriction operator: Jf = f Σ . Its adjoint J ∗ extends a function f , defined on Σ with 0 on E \ Σ. R Theorem 20. If ha+V (x)2 dm(x) < ∞ for some a > 0, then the Σ semigroup difference ˙ ˙ DΣ (t) := exp −t K0 +V − J ∗ exp −t K0 +V J, t > 0, Σ consists of Hilbert-Schmidt operators. Here Jf = f Σ . Motivation. By Weyl’s theorem we get: if DΣ (t) is Hilbert-Schmidt, then ˙ ˙ . = σess K0 +V σess K0 +V Σ R Theorem 21. If ha+V (x)1/2 dm(x) < ∞ for some a > 0, then the Σ operators DΣ (t), t > 0, are trace (i.e. in the trace class). Motivation. If DΣ (t) is trace, then ˙ ˙ σac K0 +V = σac K0 +V , and the wave operators Σ ∗ ˙ ˙ Ω± = s- lim exp ±it K0 +V J exp ∓it K0 +V Σ

t→∞

2 ˙ ˙ L exist and are unitary from Pac K0 +V L2 (Σ, m) onto Pac K0 +V Σ (E, m). In the proof of Theorem 21 the following result on trace operators may be used: see Demuth, Stollmann, Stolz, Van Casteren [13].

Lemma 22. Let K1 and K2 be integral operators with kernels k1 (x, y) and k2 (x, y) respectively. Suppose that the integral sZ Z Z dz |k2 (x, z)|2 dx |k1 (z, x)|2 dx is finite. Then K2 ◦ K1 is a trace operator and its trace norm is dominated by the latter integral.


261

A formal, but not necessarily rigorous, proof of Lemma 22 reads as follows: Let U be partial isometry for which |K2 ◦ K1 | = U K2 ◦ K1 . Then, loosely speaking, Z Z kK2 ◦ K1 ktrace = dx dz [U k2 (·, z)] (x)k1 (z, x) Z Z (4.1) = dz dx [U k2 (·, z)] (x)k1 (z, x). The expression in 4.1 is less than or equal to the integral in Lemma 22. Put ˙ ˙ (4.2) − exp −t K0 +W D(t) = exp −t K0 +V and let D(t, x, y) be its integral kernel. Put Z t A(t) = (W (X(u)) − V (X(u))) du 0

and Vs = (1 − s)V + sW. Inequality (4.3) is used in the proof of Theorem 24. Lemma 23. The following identity and inequality are true: Z t Z Z 1 D(t, x, y) = A(t) ds exp − Vs (X(u))du dµt,y 0,x ; 0

0

(4.3) |D(t, x, y)| ≤

Z

2

A(t)

dµt,y 0,x

Z Z 0

1

1/2

!1/2 Z t 2 ds exp − Vs (X(u))du dµt,y . 0,x 0

Theorem 24. R (a) If dxEx A(t)2 < ∞, then D(t) is Hilbert-Schmidt. Rp (b) If Ex [A(t)2 ]dx < ∞, then D(t) is a trace operator. Remark. The latter result is probably also true if A(t) is of the form: (4.4) Z t Z Z t i t A(t) = −i a(b(s).db(s) − div(a(b(s)))ds − V (b(u))du, 2 0 0 0

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corresponding to a particle under the influence of a magnetic field with vector potential a with Hamiltonian: 1 H(a, V ) = (−i∇ − a)2 + V. 2 R p So exp (−tH(a, V )) − exp (−tH(0, 0)) is a trace operator if dx Ex[A(t)2] is finite. The proof of the following theorem can be found in [12], and also in [13]. Theorem 25 Suppose that the unperturbed semigroup exp (−tK0 ) : t ≥ 0 is L1 -L∞ -smoothing. Then the following assertions hold true. (a) Suppose W − V belongs to L1 (E, m). Then the wave operators ˙ ˙ Ω± = s- lim exp ±is K0 +V exp ∓is K0 +W s→∞

˙ ) and they are complete. exist on the range of Pac (K0 +W 1 (b) W − V ∈ L (E, m) does not imply that D(t) is a trace operator. (c) Suppose that the function ha+V belongs to L1 (E, m). Then the Σ wave operators ˙ ˙ +V ΩΣ,± := s- lim exp ±is K0 +V J exp ∓is K 0 Σ Σ s→∞

exist and are complete. The operator JΣ restricts a function to Σ. The proof is similar to the previous one. It uses a singular version of Duhamel’s formula: see Corollary 18. A proof of the results in (1) through (8) can be found in [15]. Throughout it is assumed that the unperturbed semigroup {exp (−tK0 ) : t ≥ 0} is L1 -L∞ -smoothing. Summary. 1. exp − 2t K0 |W − V |2 ∈ L1/2 (E, m) implies: D(t), t > 0, is a trace operator; 2. W − V ∈ L2 (E, m) implies: D(t), t > 0, is a Hilbert-Schmidt operator; 3. W − V ∈ L1 (E, m) implies that the wave operators exist and are complete; 4. ha+V ∈ L1/2 (E, m) implies: DΣ (t), t > 0, is a trace operator; Σ 5. ha+V ∈ L2 (E, m) implies: DΣ (t), t > 0, is a Hilbert-Schmidt Σ operator;


263

6. ha+V ∈ L1 (E, m) implies that the wave operators exist and are Σ complete; 7. W − V ∈ C0 (E) implies: D(t), t > 0, is compact; 8. ha+V ∈ C0 (E) implies DΣ (t), t > 0, is compact; Σ 9. In 3, 4 and 6 (extensions) of the classical Pearson estimates are available. We owe this result to Demuth and Eder [12]. Problem 5. What happens to the results in the above summary Rt if the additive functional 0 V (X(u))du is replaced with more general additive processes, like stochastic integrals? The general result on the existence of Feynman-Kac semigroups can be formulated and proved for a Kato type class of additive processes. This result was obtained by L. Smits (Antwerp). The compactness properties for these more general processes are not yet investigated in its full generality. Problem 6. What happens to the compactness results, if the Dirichlet type boundary condition is replaced with a Neumann type boundary condition? Next we consider self-adjoint operators Hj = Hj∗ ≥ −ωj I, where ωj > −∞, (Hamiltonians) in the respective Hilbert space Hj , j = 0, 1. Let Vj (t) = exp (−tHj ), t ≥ 0, be the strongly continuous semigroup generated by Hj , j = 0, 1. Let J : H1 → H0 be a continuous linear operator. It is considered as an identification operator. Furthermore, let Ψ : R → C and Φ : R → R be Borel measurable functions with the following properties (the operators Φ (K0 ), Ψ (K0 ), and Ψ (H1 ) are defined via spectral theory and symbolic calculus): 1. The operators Ψ (Hj ) : Hj → Hj , j = 0, 1, are continuous, and H1 H1 Ψ (H1 ) H1,ac is dense in H1,ac . 2. There exists an admissible function α : R → R such that α(Φ(Hj )) = Hj , j = 0, 1. 3. The operator Ψ (K0 ) (Φ (K0 ) J − JΦ (H1 )) Ψ (H1 ) is of trace class. 4. The operator (Ψ (K0 ) J − JΨ (H1 )) Ψ (H1 ) is compact. Here a real-valued function α is said to be admissible if there exists a sequence of open, mutually disjoint, intervals (In : n ∈ N) in R such that 1. the function α is continuously differentiable on R; 2. α0 (x) > 0, x ∈ R; S 0 3. on each closed sub-interval of ∞ n=1 In the function α is of bounded variation.

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Let E1 (·) be the spectral decomposition of the operator H1 . A vector H1 if the measure B 7→ hE (B)g, gi is absolutely cong belongs to Ha,ac 1 tinuous with respect to the Lebesgue measure. In 3 we may take H0 = H1 = L2 (E, m), J = I, Ψ(λ) = e−λt0 , Φ(λ) = λ (and hence α(λ) = λ), ˙ , H1 = K0 +W ˙ . In 4 we may take H0 = L2 (Σ, m), K0 = K0 +V 2 H1 = L (E, m), Jf = f restricted to Σ, Ψ(λ) =Φ(λ) = α(λ) = λ, ˙ ˙ H0 = exp −t0 K0 +V , H1 = exp −t0 K0 +V . In 6 we may take Σ 2 2 H0 = L (Σ, m), H1 = L (E, m), Jf = f restricted to Σ, Ψ(λ) = e−λt0 , ˙ . ˙ Φ(λ) = λ (and hence α(λ) = λ), H0 = K0 +V , H1 = K0 +W Σ In [12] the authors prove the following theorem. Theorem 26. Let β : R → R be any admissible function. Let g H1 belong to H1,ac . The following Pearson estimate is true:

2

2 (4.5)

(Ω± (β (H0 ) , J, β (H1 )) − J) ψ (H1 ) g 0 ≤ 16π kΨ (H0 ) JΨ (H1 )k ( 1 , 0 ) kΨ (H0 ) (Φ (H0 ) J − JΦ (H1 )) Ψ (H1 )ktrace( + k(Ψ (H0 ) J − JΨ (H1 )) Ψ (H1 )k2 ( 1 , 0 ) !

d

× kgk2 1 + .

dλ EΦ(H1 ) (−∞, λ]g, g ∞ L (R)

1.

0)

Here EΦ(H1 ) (B) = E1 Φ−1 (B) denotes the spectral decomposition of Φ (H1 ). Remark. If in (4.5) we set Ψ ≡ 1, and Φ(λ) = λ, we get (4.50 )

k(Ω± (β (H0 ) , J, β (H1 )) − J) gk2

0

≤ 16π kJk ×

( 1 , 0 ) kH0 J − JH1 ktrace( 1 . 0 ) !

d

kgk2 1 + ,

dλ EΦ(H1 ) (−∞, λ]g, g ∞ L (R)

which is slightly worse than the classical Pearson estimate: (4.500 ) 1 k(Ω± (β (H0 ) , J, β (H1 )) − J) gk2 16π

0


≤kJk

(

1,

0)

kH0 J−JH1 ktrace(

1.

265

d

E (−∞, λ]g, g

0) dλ Φ(H1 )

.

L∞ (R)

5. Some (abstract) problems related to Neumann semigroups This section is motivated by the following observation. A general motivation is the following one. Let {(Ω, F, Px ) , (X(t), t ≥ 0) , (ϑt : t ≥ 0) , (Rν , B)} be the Markov process of ν-dimensional Brownian motion. Consider a Neumann initial value problem on an open domain Σ in Rν . This means find a solution to the following Cauchy problem: ∂ 1 (5.1) u(x, t) = ∆u(x, t) on Σ, ∂t 2 where its normal derivative Dn u(x, t) = 0, x ∈ ∂Σ. Assume u(x, 0) = f (x), x ∈ Σ. Find a sequence of multiplicative processes Mn (t) such that u(x, t) = lim Ex [Mn (t)f (X(t))] . n→∞

If we consider Dirichlet semigroups, then such multiplicative functionals can be found. For instance, the semigroup in (3.1) can be written as ˙ (5.2) exp −t K0 +V f (x) = lim Ex [Mn (t)f (X(t))] , Σ

where

Mn (t) = exp −n

n→∞

Z

t

Z t 1E\Σ (X(s)) ds exp − V (X(s))ds .

0

0

We extend this kind of problem to a general domain in a locally compact −v space. Let v : E → [0, ∞) be a function R in D(K0 ) with e ∈ D(K0 ) as well. We suppose that K0 1 = 0, i.e. K0 f (x)dx = 0, for f ∈ D (K0 ), K0 f ∈ L1 (E, m). Put Z t (5.3) Mv (t) = v(X(t)) − v(X(0)) + K0 v(X(s))ds; 0

(5.4) hMv i (t) = hMv , Mv i (t) =

Z

t

Γ1 (v, v) (X(s))ds;

0

Zv (t) = M−v (t) − (5.5)

1 hMv i (t) 2

= v(X(0)) − v(X(t)) +

Z 0

t

ev(X(s)) K0 e−v (X(s))ds;

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Jan A. Van Casteren

(5.6) Z t Tv (t)f (x) = Ex exp Zv (t) − V (X(s))ds f (X(t)) 0 (5.7) = Ex exp (v(X(0)) − v(X(t))) Z t v(X(s)) −v exp e K0 e (X(s))ds 0 Z t × exp − (5.8) V (X(s))ds f (X(t)) 0

(5.9) (11)

= [exp (−tKv ) f ] (x), where Kv f = ev K0 e−v f + V − ev K0 e−v f.

In particular Kv 1 = V . Here Γ1 (f, g) is the carré du champ operator, introduced by Roth [37], but popularized by Bakry (see e.g. [4]): (5.10) 1 Γ1 (f, g)(x) = lim Ex [(f (X(s)) − f (X(0))) (g(X(s)) − g(X(0)))] , s↓0 s and hMv i = hMv , Mv i is the variation process corresponding to the martingale Mv . The family {Tv (t) : t ≥ 0} is a strongly continuous semigroup in L2 (E, m). Put u(t, x) = Tv (t)f (x). Then u(0, x) = f (x) and ∂u = −K0 u − V u − Γ1 (v, u). ∂t So the expression f 7→ Γ1 (v, f ) is sort of a drift in gradient form. The corresponding quadratic form is given by (5.11)

EVv (f, g) Z Z = − vΓ1 (f, g)(x)dx + 2v(x) + 1 K0 f (x)g(x)dx Z + V (x)f (x)g(x)dx.

Notice the identity K0 (f g) + Γ1 (f, g) = (K0 f )g + f (K0 g). Put Pv,x (A) = Ex exp (v(X(0)) − v(X(t))) Z t v(X(s)) −v exp e K0 e (X(s))ds 0 Z t (5.12) × exp − V (X(s))ds 1A , 0


267

where A belongs to Ft . In case K0 is − 21 ∆, we have Γ1 (f, g) = ∇f · ∇g, and Mv (t) =

Z

t

∇v(X(s))dX(s)

0

(Itô integral), where X is Brownian motion. In addition, (5.13) Ex [F (Y (s) : 0 ≤ s ≤ t) exp (Zv (t))] = Ex [F (X(s) : 0 ≤ s ≤ t)] , Rs where Y (s) = X(s) + 0 ∇v(X(σ))dσ. This is a version of the Girsanov transformation. For an up-to-date account of Girsanov transformations, ¨ unel and Zakai [43]. We want to take (sinthe reader is referred to Ust¨ gular) limits in the expressions for Tv (t) and EVv , for v tending to 1E\Σ , 0 < a ≤ ∞. For a < ∞, the quadratic form converges to Z Z V Γ1 (f, g)(x)dx + 2a (5.14) EΣ,a (f, g) := − a K0 f (x)g(x)dx E\Σ E\Σ Z Z + K0 f (x)g(x)dx + V (x)f (x)g(x)dx. So that, if

R

E\Σ (2K0 f (x)g(x)

− Γ1 (f, g) (x)) dx = 0, then

(5.15) EVΣ,N (f, g)

:=

lim EV (f, g) a→∞ Σ,a

=

Z

K0 f (x)g(x)dx +

Z

V (x)f (x)g(x)dx.

Here EVΣ,N should stand for Neumann quadratic form. In the presence of the carré du champ operator we may define a distance on E × E: d(x, y) = sup {|ψ(y) − ψ(x)| : Γ1 (ψ, ψ) ≤ 1} . The local time (occupation) the process X (up to time t) spends on the (boundary of the) complement of Σ is then the bounded variation part of the process d (X(t), E \ Σ). Suitable logarithmic Sobolev inequalities imply d(x, y) < ∞: see e.g. Bakry [4], Théorème 3.2, page 39. A proof of the following result may be based on § 3 of Bakry [4] in combination with the proof of Lemma 3.2.1 in Davies [11], p. 83. A detailed proof can be found in [15] Chapter 1, §D. Another somewhat less general result, but with a simple proof, is to be found in Léandre [29].

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Theorem 27. Suppose that there exists a continuous function m : (0, ∞) → (0, ∞) with the property that for every 1 < u < ∞ the following logarithmic Sobolev inequality is true: (5.16)

E2 (f ) ≤ uE2 (f ) + m(u) kf k22

for all f > 0, fR ∈ A. Let c(u) > 0 be a function in L1 ([1, ∞)) with the ∞ property that 1 c(u)du = 1. The quantity mc (t) is defined by Z ∞ m 4tc(u)(u − 1) (5.17) du. mc (t) = u2 1 (1) Let t > 0 be such that the corresponding quantity mc (t) is finite. Then exp (−tK0 ) maps L1 (E, m) to L∞ (E, m) and kexp (−tK0 )k∞,1 ≤ exp (mc (t)) . (2) Put 1 η = ηc = 2

Z 1

∞

1 c(u) u − 1 + − 2 du u−1

and suppose that η is finite. Let ψ be a function in A with the property that Γ1 (ψ, ψ) ≤ 1. Suppose that mc (t/2) is finite. Then ! |ψ(y) − ψ(x)|2 (5.18) exp (−tK0 ) (x, y) ≤ exp (mc (t/2)) exp − . 2t(1 + η) The quantities Ep (f ) (“entropy”) and Ep (f ) (“energy”), 1 ≤ p < ∞, f > 0, are defined via the following formulae: ! Z f (x)p p (5.19) Ep (f ) = f (x) log dx; kf kpp E

(5.20) Ep (f ) = K0 f, f p−1 , f ∈ D (K0 ) . For p = 2 these expressions also make sense for complex-valued functions f ∈ D (K0 ). R∞ Remark 1. Let c(u) be as in the theorem and suppose that 2 c(u)du = 1/2. Define the function c1 (v), v > 1, by  v 1 c (v) = c , if 1 < v ≤ 2; 1 v − 1 (v − 1)2  c1 (v) = c(v), if v ≥ 2. Then the hypotheses and conclusions of Theorem 27 remain valid with c1 (u) instead of c(u).


269

Remark 2. The operator K0 generates a diffusion in the following sense: for every C ∞ -function Φ : Rν → R, with Φ(0, . . . , 0) = 0, the following identity is valid: (5.21) K0 (Φ(f1 , . . . , fn )) =

n n X ∂Φ 1 X ∂2Φ (f1 , . . . , fn ) K0 fj − (f1 , . . . , fn ) Γ1 (fj , fk ) ∂xj 2 ∂xj ∂xk j=1

j,k=1

for all functions f1 , . . . , fn in a rich enough algebra of functions A, contained in the domain of the generator K0 , as described in Remark 3. Remark 3. The algebra A in Theorem 27 has to be “large” enough. To be specific, it is supposed to possess the following properties: It is dense in Lp (E, m) for all 1 ≤ p < ∞ and it is a core for K0 considered as an operator in L2 (E, m). In addition, it is assumed that A is stable under composition with C ∞ -functions of several variables, that vanish at the origin. Moreover, in order to obtain some nice results a rather technical condition is required: whenever (fn : n ∈ N) is a sequence in A that converges to f with respect to the graph norm of K0 (in L2 (E, m)) and whenever Φ : R → R is a C ∞ -function, vanishing at 0, with bounded derivatives of all orders (including the order 0), then one may extract a subsequence (Φ (fnk ) : k ∈ N) that converges to Φ(f ) in L1 (E, m), whereas the sequence (K0 Φ (fnk ) : k ∈ N) converges in L1 (E, m) to K0 Φ(f ). Notice that all functions of the form eψ f , ψ, f ∈ A, belong to A. This fact was used in the proof of Theorem 27. Also notice that the required properties of A depend on the generator K0 . In fact we will assume that the algebra A is also large enough for all operators of the form f 7→ e−ψ K0 eψ f , where ψ belongs to A. Roughly speaking the problem can be described as follows: Problem 7. What relations, if any, do exist between the following concepts: 1. singular limit of quadratic form; 2. singular limit of Feynman-Kac semigroup; 3. local time spent by the process X in E \ Σ; 4. Girsanov transformation (SDE); 5. reflected Markov process? Problem 8. A related, not completely understood problem, is to formulate and prove the precise correspondence between convergence of

270

Jan A. Van Casteren

semigroups and of the associated quadratic forms. One ought to consider Γ-convergence of quadratic forms. For the latter see e.g. Dal Maso [32]. Problem 9. If possible, incorpurate Neumann scattering in our discussion. Problem 10. Suppose the sequence {vn : n ∈ N} converges to a1E\Σ in a reasonable way (a = 12 , or, more generally, ∞ ≥ a > 0). Does it follow that the sequence (Pvn ,x ) converges or is tight? Instead of a genuine drift we might also consider an imaginary drift term: ∂u (5.22) = −K0 u + i(K0 v)u − V u − iΓ1 (v, u), u(0, x) = f (x). ∂t A solution to equation (5.22) is given by the Feynman-Kac formula: u(t, x) = Ex [exp (A(t)) f (X(t))] ,

where Z 1 t A(t) = −iv(X(t)) + iv(X(0)) + Γ1 (v, v) (X(s))ds. 2 0

Problem 11. What happens to this if we take singular limits, i.e. if we let v tend to a1E\Σ , 0 < a ≤ ∞? Problem 12. Let ha+V be an a+V -harmonic function with “normal Σ, derivatives” equal to 1. Is the following conjecture true? Conjecture 28. (a) If ha+V belongs to L2 (E, m), then DΣ, (t), t > 0, are HilbertΣ, Schmidt operators. 1/2 (b) If ha+V belongs to L1 (E, m), then DΣ, (t), t > 0, are trace Σ, class operators. There exist papers related to the problems which we presented above. There is one by Williams et Zheng [50], where reflected Brownian motion is constructed as a limit in law of processes, with a strong drift close to the boundary. Another related paper is [36] by Pardoux and Williams. An older paper is one on one-dimensional stochastic differential equations involving local times by J.-F. Le Gall [22]: see the remark on page 72/73. Concluding remarks. In this paper we proved some theorems about the close connection that exists between probability theory and


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analysis. We mentioned some (open) problems in connection with Neumann type semigroups. We hope that the readers are inspired by some of the problems and results which are presented. It presents some rather general techniques (Markov processes, Feynman-Kac formula, martingale theory, squared gradient operator, classical harmonic analysis) to prove results in operator theory: unique Markov extension, some results in connection with scattering and spectral theory, a result on heat kernel estimate, compactness properties of differences of self-adjoint semigroups.

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Department of Mathematics and Computer Science University of Antwerp (UIA) Universiteitsplein 1 2610 Antwerp/Wilrijk, Belgium E-mail : [email protected]