arXiv:1405.2487v3 [math.PR] 1 Mar 2016
Global limit theorems on the convergence of multidimensional random walks to stable processes. A. Agbor ∗, S. Molchanov †, B. Vainberg
‡
Abstract Symmetric heavily tailed random walks on Z d , d ≥ 1, are considered. Under appropriate regularity conditions on the tails of the jump distributions, global (i.e., uniform in x, t, |x| + t → ∞,) asymptotic behavior of the transition probability p(t, 0, x) is obtained. The examples indicate that the regularity conditions are essential.
Key words: random walk, heavy tail, stable law, large deviations, global asymptotics
1
Introduction
The paper is motivated by two applied problems from the theory of homopolymers and from branching diffusion processes. Both problems can be reduced to asymptotic analysis of solutions to parabolic Schr¨odinger type equation ut = −Hu := Lu + V (x)u,
u(0, x) = u0 (x).
(1)
Here t ≥ 0, x ∈ Z d (other phase spaces X can be considered, but we will restrict ourselves to the lattice case in this paper), L = L∗ is a self-adjoint non-positive operator on L2 (Z d ) generating the underlying random walk, and V is a potential. In the case of homopolymers, problem (1) with a fast decaying (or even compactly supported) potential V (x) ≥ 0 describes the shape of a long polymer chain in the field of an attracting potential (see [2], [3], [4] and references there). In the case of branching diffusion processes, V (x) is proportional to the rate of birth of new particles. There are two simplest possibilities: V (x) = β is a constant or V (x) ≥ 0 is compactly supported. In the first case, the rate does not depend on x. In particular, this problem appears in the classical model of population introduced (when X = Rd ) by ∗
Dept of Mathematics and Statistics, UNC at Charlotte, NC 28223,
[email protected]. Dept of Mathematics and Statistics, UNC at Charlotte, NC 28223,
[email protected]. ‡ Dept of Mathematics and Statistics, UNC at Charlotte, NC 28223,
[email protected]; corresponding author. †
1
Kolmogorov, Petrovskii and Piskunov. In the second case, the new particles are produced in a bounded region, see [14], [15]. In all the publications mentioned above, as well as in many others, the generator L is local. Usually, it is the Laplacian (when X = Rd ) or lattice Laplacian. However, some more profound models of population dynamics were introduced recently in mathematical biology where the underlying brownian motion in Rd (or simple symmetric random walk on Z d ) is replaced by a heavy- tailed Markov process (containing long jumps, the so-called Levy flights). The main models of this type are contact processes introduced in [10], [9]. The main goal of the present paper is a study of the global asymptotic behavior of the transition probability for the random walk (time is continuous) with heavy tails. We consider below only lattice models, but it is not difficult to carry the results over to the continuous case. Let us describe the model under consideration. Let x(t) be the random walk on Z d with continuous time and the generator X Lf (x) = κ (f (x + z) − f (x))a(z), x ∈ Z d , z∈Z d
P where a(z) = a(−z) ≥ 0 (we assume that a is symmetric), z∈Z d a(z) = 1, κ > 0. The process spends an exponentially distributed time τx at each site x ∈ Z d (P {τx > s} = exp(−κs)) and jumps at time τx + 0 from x to x + z with the probability a(z). The transition probabilities p(t, x) = Px {x(t) = x; x(0) = 0} satisfy the equation pt = Lp,
p(0, x) = δ(x).
After the Fourier transform, it takes the form pbt (k) = b a(k)b p(k), pb(0, k) = 1, where b a(k) is a periodic function given by X X b a(k) = e−i(k,z) a(z) = cos(k, z)a(z). (2) z∈Z d
z∈Z d
Thus
Z 1 ei(k,x)+κt(ba(k)−1) dk, T d = [−π, π]d . p(t, x) = d (2π) T d If L is the Laplacian, then a(z) = 1/2d when |z| = 1 and a(z) = 0 when |z| = 6 1. In our case, a(z) does not have a compact support, and moreover, we assume that it decays slowly at infinity. P If E|x(t)|2 < ∞, i.e., z∈Z d |z|2 a(z) < ∞, then the local (|x| = O(t1/2 )) limit theorem holds [7] (under minimal additional assumptions on x(t)): P {x(t) = x} ∼
e(B
−1 x,x)/2tκ2
√ , (2πtκ2 )d/2 detB
x ∈ Z d,
|x| ≤ Ct1/2 ,
where matrix B is the covariance of the jump distribution a(z): B = [−
∂2b a(k) ]|k=0. ∂ki ∂kj 2
t → ∞,
This local limit theorems (in the central zone) were extended [7], [8] for a wide class of stationary and ergodic processes with fast decaying correlations. We assume that the second moment does not exists, but function a(z) has a regular behavior at infinity: a(z) =
a0 (z) ˙ (1 + o(1)), d+α |z|
|z| → ∞,
z˙ =
From here it follows that
z , a0 (z) ˙ > δ > 0, |z|
α ˙ b a(k) − 1 = −b0 (k)|k| (1 + o(1)),
|k| → 0,
0 < α < 2.
k k˙ = , |k|
(3)
(4)
˙ is an appropriate integral transformation of a0 (z). where b0 (k) ˙ Relation (4) can be proved under a weaker assumption than (3). We assumed that a(−z) = a(z) (i.e, the random walk is symmetric). In this case, Z απ ˙ ˙ α dSx˙ > 0, b0 (k) = −Γ(−α) cos a0 (x)|( ˙ x, ˙ k)| (5) 2 S d−1 where Γ is the gamma-function. Then a different local limit theorem holds: P {x(t) = x} ∼
1 td/α
S(
x t1/α
),
x ∈ Z d,
|x| ≤ Ct1/α ,
t → ∞,
(6)
where S(y) = Sα,a0 (y) is the density of the stable d-dimensional law depending on α ∈ (0, 2) and function a0 in (3). Function S is given by its Fourier transform (characteristic function): α ˙ b S(k) = e−b0 (k)|k| , k ∈ Rd .
The local limit theorem (6) in the case of d = 1 can be found in [7], and it can be proved similarly for arbitrary d. Recall (see [5, Ch XII, §11]) that the most general symmetric stable law has the characteristic function R α ˙ α ˙ ψα,µ (k) = e−|k| S d−1 |(x,˙ k)| µ(dx) ,
where µ(dx) ˙ is a symmetric (with respect to the reflection over the origin) finite measure d−1 on S . In our case, the measure has the density ca0 (x) ˙ ≥ δ > 0. The goal of this paper is to establish the global asymptotic behavior of p(t, x) = P {x(t) = x} when t + |x| → ∞ without restrictions on the relations between t and |x|. Thus the zone of large deviations (|x| = O(t1/α+δ ) or even |x| = O(eδt )) is also included. One can’t expect essential results in this direction without strong additional assumptions on the tails of a(z) that lead to a more specific behavior of b a(k) as k → 0. Namely we assume that a(z) =
d+ǫ X aj (z) ˙ 1 + O( ), d+α+j 2d+α+1+ǫ |z| |z| j=0
|z| → ∞,
α ∈ (0, 2),
aj ∈ C d+1−j+ǫ (S d−1 ), (7)
3
where ǫ = 1 if α = 1 and ǫ = 0 otherwise, and a0 (z) ˙ > δ > 0. The following lemma (which will be proved in Appendix) provides the asymptotic behavior of b a(k) at zero. Lemma 1.1. If (7) holds, then b a(k) = 1 −
d X
α+j ˙ bj (k)|k| + f (k),
j=0
k ∈ T d = [−π, π]d ,
f (0) = 0,
(8)
where bj ∈ C d+[α]+1 (S d−1 ) and function f , being extended periodically on Rd , belongs to C d+[α]+1 (Rd ). α ˙ Moreover, the homogeneous function −b0 (k)|k| in Rd is the Fourier transform of the −d−α homogeneous (of order −d−α) distribution that is equal to a0 (x)|x| ˙ when 0 6= x ∈ Rd , and Z απ ˙ ˙ α dSx˙ > 0, b0 (k) = −Γ(−α) cos a0 (x)|( ˙ x, ˙ k)| (9) 2 S d−1 where Γ is the gamma-function. Remarks. 1) Note that f can not be omitted, since a change in the values of a(z) at several points does not perturb its asymptotic behavior at infinity, but changes b a(k) by an analytic function. 2) The next two properties of b a(k) follow immediately from properties of a0 (x): ˙
b a(k) < 1, 0 6= k ∈ T d . (10) P The latter inequality follows from (2) and the assumption x∈Z d a(x) = 1 if, for each k ∈ T d , k 6= 0, there is a point z ∈ Z d , where e−i(x,k) 6= 1 and a(x) 6= 0. Such points z exist due to (7). The following uniform asymptotics of the function p(t, x) = p(t, x, 0) is one of the main results of the present paper. We put κ = 1 everywhere below, since this condition can be satisfied after simple substitution κt → t. b a(−k) = b a(k);
Theorem 1.2. (1) Let (8)-(10) hold. Then p(t, x) = where
1 td/α
S(
x t1/α
)(1 + o(1)),
1 S(y) = (2π)d
Z
Rd
when ˙
x ∈ Z d,
|x| + t → ∞,
α
ei(k,y)−b0 (k)|k| dk > 0, y ∈ Z d ,
is the stable density S = Sα,a0 (y). |x| (2) If t1/α → ∞, |x| ≥ 1, then the previous statement can be specified as follows: p(t, x) =
a0 (x)t ˙ a0 (x) ˙ t1/α d+α ( ) (1 + o(1)) = (1 + o(1)). d/α d+α t |x| |x| 4
(11)
Remark 1. The study of global limit theorems for the sums sn = x1 + ... + xn of i.i.d.r.v. was initiated by Yu. Linnik (see [7], Ch XIV). In 1-D case, he proved a uniform in x ∈ R1 local limit theorem for the density pn (·) of sn under regularity conditions on the tails similar to (7). Some extensions of these results (based on the technique of quasi-cumulants) can be found in [13], [16]. Remark 2. Our proof allows one to write the next terms of the global asymptotics of p(t, x) under additional assumptions on a(x) or b a(k) (more terms in the asymptotics (7), (8) and additional smoothness). The regularity condition (7) on the tails of a(x) usually holds in applications. Therefore, (8)-(10) also hold. One has to be careful and check the validity of the latter relations if (7) is violated. The following examples show that the smoothness assumptions in (8) are essential. Example 1. The first example concerns a random variable in R3 , but it also will be used to construct a specific random walk on Z 3 in Example 2. 4r Consider a random variable with an isotropic distribution density α(r) = sin in π2 r4 3 R , r = |x|. Then Z Z ∞Z 4π ∞ 2 −ikx r sin(|k|r)α(r)dr, (12) α b(k) = r e α(r)dSdr = |k| 0 0 S2
where S 2 is the unit sphere in R3 . We put sin4 r = 38 − 21 cos(2r) + 81 cos(4r) in the expression for α(r) and evaluate the integral using the following classical identity: Z ∞ sin(ar) π dr = sgna. r 2 0 This leads to
3|k| |k| ≤ 2 1 − 8 , |k| 2 α b(k) = |k| − 1 + 8 , |k| ∈ [2, 4] 0, |k| > 4.
˙ = 3/8, and bj (k) ˙ = 0, j > 0, but Function α b(k) has the form (8) with α = 1, b0 (k) function f (k) for α b(k) in our example is not as smooth as Theorem 1.2 requires: it belongs to C 1 (R3 ), but already does not belong to C 2 (R3 ). R 1 One can evaluate the transition density p(t, x) = (2π) et(bα(k)−1)+ixk dk by using 3 R3 spherical coordinates, evaluating the integral over the sphere S 2 as in (12), followed by integration by parts three times in |k|. This leads to p(t, x) =
t π2r4
3 3 1 t 1 [ − cos(2r)e− 4 t + cos(4r)e−t ] + O( 5 ), 8 2 8 r
r ≫ t.
This oscillatory behaviour of p is different from the one stated in Theorem 1.2. Example 2. Using the function α b(k) defined in the previous example and the Poisson summation formula, one can construct the characteristic function b a(k) of the lattice 5
distribution: b a(k) =
X
n∈Z d
α b(k + 2πn).
Since α b(k) = 0 for |k| > 4, we have b a(k) = α b(k) on T d = [−π, π]d . Thus the jump distribution a(x) and the transition probabilities p(t, x) coincide with the restrictions on the lattice Z d of the corresponding expressions in Example 1. Example 3. Consider the following well known example: the stable distribution given by the characteristic function Pd α b S(k) = e− j=1 |kj | . Then
b a(k) = 1 − |k|
α
d X j=1
|k˙ j |α + O(|k|1+α),
˙ is not smooth. Since S(x) = i.e., function b0 (k) infinity depends significantly on x. ˙ Say, S(r, 0, ..., 0) ∼
c r 1+α
but
Qd
k → 0,
j=1 Sα (xj ),
,
the asymptotics of S(x) at
r → ∞,
cd
, r → ∞. r d(1+α) Note that the problem of the asymptotic behaviour for general stable distributions is not simple. There are many relatively recent publications on that topic, see [1], [6], [17] and references there. S(r, r, ..., r) ∼
2
Proof of the main result
The proof will use three lemmas proved below and the following simple facts. Since function b a(k) is even, we have ∇f (k) = 0 at k = 0.
(13)
From (4), (9) and (10) it follows that b a(k) − 1 < −γ|k|α ,
k ∈ [−π, π]d ,
γ > 0.
(14)
Let ψ = ψ(τ ) ∈ C ∞ (R1 ) be a cut-off function such that ψ(τ ) = 1, |τ | < 1, ψ(τ ) = 0, |τ | > 2. Let Z I=
[−π,π]d
ei(k,x)+t(ba(k)−1) [1 − ψ(|k|t1/α )]dk.
6
Lemma 2.1. Let (7) hold. Then the following estimate is valid for all t, |x| > 0: |I| ≤
t1/α d+[α]+1 ( ) . td/α |x| C
Proof. From (14), (4) it follows that |∂kj et(ba(k)−1) |
≤C
|j| X
α
(|k|lα−|j|tl )e−γ|k| t ,
k ∈ [−π, π]d , 0 < |j| ≤ d + [α] + 1.
l=1
(15)
a(k) t(b te a(k)−1) , and the pre-exponential factor can be estimated Indeed, ∂k∂ s et(ba(k)−1) = ∂b∂k s from above by C|k|α−1 t, due to (4), (13). Each next differentiation leads to the appearance of an additional similar pre-exponential factor (when the derivative is applied to the exponent) or to decreasing of the power of k in the estimate of the pre-exponent by one. These arguments justify (15). α Since |k|lα tl e−γ|k| t/2 ≤ C, estimate (15) implies that
|∂kj et(ba(k)−1) | ≤ Ct|j|/αe−γ|k|
α t/2
|k|t1/α > 1, |j| ≤ d + [α] + 1.
,
Hence, for each s = 1, 2, .., d and m = d + [α] + 1, we have |
∂m α [et(ba(k)−1) (1 − ψ(|k|t1/α ))]| ≤ Ctm/α e−γ|k| t/2 , m (∂ks )
and therefore (since b a(k) is periodic), Z ∂m −m ei(k,x) |I| = |(−ixs ) [et(ba(k)−1) (1 − ψ(|k|t1/α ))]dk| m (∂ks ) [−π,π]d Z Ctm/α C1 t(m−d)/α −γ|k|α t/2 ≤ e dk = . |xs |m Rd |xs |m This completes the proof of the lemma, since s is arbitrary. Let
1 I1 = (2π)d
Z
ei(k,x)+t(ba(k)−1) ψ(|k|t1/α )dk.
|k|t1/α 0, |x| > rt1/α and |x| + t → ∞, the |x| →∞ following asymptotics holds when t1/α I1 =
a0 (x) ˙ t1/α d+α a0 (x)t ˙ ( ) (1 + o(1)) = (1 + o(1)), td/α |x| |x|d+α
where there exist ε > 0 and for each r > 0 there exists C = C(r) such that |o(1)| < C(|x|−ε + (
t1/α ε ) ), |x| 7
|x| > rt1/α , |x| > 1.
Proof. The substitution kt1/α → k followed by integration by parts implies that, for each N > 0, Z t t1/α N −d−α t 1 t1/α N ) = CN d+α ( ) = d+α o(1). | ei(k,x) ψ(|k|t1/α )dk| < CN d/α ( t |x| |x| |x| |x| |k|t1/α rt1/α ), and therefore the last two integrals can be estimated by C(t|x|−d−α−δ + t2 |x|−d−2α ) =
t o(1). |x|d+α
Now it remains to prove the statement of the lemma for the integral Z −t α α ˙ ˙ I3 = ei(k,x) b0 (k)|k| ψ(|k|t1/α )dk = tF −1 (−b0 (k)|k| ψ(|k|t1/α )). d (2π) |k|t1/α 0. Let |y| < R be the largest ball where S(y) > 0 for all y inside the ball. We need to show that R = ∞. α ˙ Consider the convolution S2 = S ∗ S. Its Fourier transform is equal to e−2b0 (k)|k| , and therefore Z d 1 1 α ˙ ei(k,x)−2b0 (k)|k| dk = 2− α S(2 α y). S2 (y) = (18) d (2π) Rd The last equality (self-similarity) can be obtained by substitution k → 21/α k. Since S(y) ≥ 0, the convolution is positive in the ball |y| < 2R. Hence (18) implies that 1 S(y) > 0 in the ball |y| < 21− α R. The radius of the latter ball is larger than R if α ∈ (1, 2), R < ∞. Thus R = ∞, i.e., S(y) > 0 everywhere if α ∈ (1, 2). Let now 0 < α ≤ 1. By passing in (11) to spherical coordinates and integrating by parts, we obtain that Z Z ∞ 1 α ˙ ˙ y)|]e S(y) = cos[|k||y||(k, ˙ −b0 (k)|k| d|k|dSk˙ d (2π) S d−1 0 Z Z ∞ ˙ y)|] sin[|k||y||(k, ˙ 1 α ˙ α−1 −b0 (k)|k| ˙ b0 (k)α|k| e d|k|dSk˙ . = d ˙ (2π) S d−1 0 |y|(k, y) ˙ ˙
α
The last integral is positive since the function b0 |k|α−1 e−b0 (k)|k| is monotone in |k|, and the integral Z ∞ sin(γτ )f (τ )dτ 0
is positive for every γ > 0 and every monotonically decreasing function f . Proof of Theorem 1.2. The second statement of the theorem follows immediately from Lemmas 2.1, 2.2. It remains to prove the first statement. First, let us prove that the second statement of the theorem is valid for the function Z 1 x 1 α ˙ ei(k,x)−tb0 (k)|k| dk = d/α S( 1/α ). p0 (t, x) = d (2π) Rd t t We split p0 in two parts p0 (t, x) = p0,1 + p0,2 , where Z Z 1 1 α α ˙ ˙ i(k,x)−b0 (k)|k| 1/α p0,1 = e ψ(kt )dk, p0,2 = ei(k,x)−b0 (k)|k| (1−ψ(kt1/α ))dk, d d (2π) Rd (2π) Rd The estimate for I, established in Lemma 2.1, is valid for p0,2 . We can not formally refer to Lemma 2.1 (since the integrals are different), but the same arguments can be applied. 9
In fact, they are simpler in the case of p0,2 . We change the variable k → kt−1/α and then integrate by parts d + [α] + 1 times with respect to each of the variables kl , 1 ≤ l ≤ d. This leads to |p0,2 | ≤
t1/α d+[α]+1 t t1/α [α]+1−α t ( ) = C ( ) = d+α o(1), d/α d+α t |x| |x| |x| |x| C
|x| α ˙ → ∞. Lemma 2.2 is valid when b a(k) − 1 = −b0 (k)|k| and where |o(1)| → 0 as t1/α can be applied to p0,1 . Together with the estimate for p0,2 this justifies that the second statement of the theorem is valid for p0 . Thus p0 and p are close to the same non-zero a(x)t ˙ function |x| d+α , and therefore are close to each other. Hence, for each ε > 0, there exists an A = A(ε) such that
|p(t, x) − p0 (t, x)| ≤ εp0 (t, x) when
|x| ≥ A, |x| ≥ A. t1/α
Now the theorem will be proved if we show that for each fixed ε, A > 0 there exists T = T (ε, A) such that |p(t, x) − p0 (t, x)| ≤ εp0 (t, x) when
|x| ≤ A, t > T. t1/α
(19)
We will show that |p(t, x) − p0 (t, x)| ≤ Ct−d/α−ε , ε > 0,
when
|x| ≤ A, t → ∞. t1/α
(20)
|x| ), where S is given by (11). Since Then we note that p0 has the form p0 (t, x) = t−d/α S( t1/α ˙ is even, S is real-valued. Since S = S(y) is continuous in y and does the function b0 (k) not vanish (see Lemma 2.3 ), p0 > c0 t−d/α > 0 when |y| ≤ A. Thus (20) implies (19), so it is enough to justify (20). Let Z 1 (1) ei(k,x)+t(ba(k)−1) dk, p (t, x) = d (2π) |k| 0 and a smooth v = v(z)) terms decay faster at infinity. Then from (22) it follows that Z
−d−γ −i(x,k) u(x)|x| ˙ e dx =
|x|>1
X
06=z∈Z d
e−i(z,k) [
n−1 X
vs (z)|z| ˙ −d−γ−s + O(|z|−d−γ−n )] + g(k),
s=0
(25) where vs ∈ C , v0 (z) ˙ = u(z), ˙ and g is an analytic in k function that appeared because we did not care to evaluate the contribution of the terms with small values of z and the contribution of the integral over the area in a neighborhood of the origin explicitly. We will use (25) only for γ and n such that n−s
γ + n = d + α + 1 when α ∈ (0, 2), α 6= 1; 12
γ + n = d + 3 when α = 1.
(26)
Then the sum in z on the right that contains the remainder term is m := d + [α] + 1 times differentiable in k, and therefore (25) implies that Z
−d−γ −i(x,k) u(x)|x| ˙ e dx = |x|>1
X
e−i(z,k)
n−1 X
vs (z)|z| ˙ −d−γ−s + q(k),
s=0
06=z∈Z d
q ∈ C m.
(27)
γ −d−γ ˙ The left-hand side in (27) differs from the Fourier transform c(k)|k| of u(x)|x| ˙ (see Lemma 2.4) by an analytic function. Thus (27) leads to the following expression for the term on the right with s = 0:
X
06=z∈Z d
γ ˙ e−i(z,k) u(z)|z| ˙ −d−γ = c(k)|k| −
X
e−i(z,k)
06=z∈Z d
n−1 X s=1
vs (z)|z| ˙ −d−γ−s + h(k), h ∈ C m ,
(28) ˙ where c(k) and u(z) ˙ are related by (21). From (26) and Lemma 2.4 (with n1 = n) it m follows that c ∈ C , i.e., the smoothness of functions c and h does not depend on n and γ (under condition (26)). This allows us to apply the induction on n = 1, 2, ... in formula (28), which leads to the following result: X
06=z∈Z d
−i(z,k)
e
u(z)|z| ˙
−d−γ
=
n−1 X s=0
γ−s ˙ cs (k)|k| + f (k), f ∈ C m ,
(29)
where cs ∈ C m , c0 = c is given by (21), and γ and n are related by (26). In particular, (29) holds for γ = α + j, u = aj for all the values of j from (7), and the summation of (29) in j implies the statement of the Lemma.
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