Recurrence of direct products of diffusion processes in random media having zero potentials

In this paper, we introduce an index which measures the strength of recurrence of symmetric Markov processes, and give some sufficient conditions for recurrence of direct products of symmetric diffusion processes. The index is given by the Dirichlet forms of the Markov processes. Moreover, as an application, we prove the recurrence of some multi-dimensional diffusion processes in random environments including zero potentials.


Introduction
Global properties of stochastic processes as well as related problems are important topics in both probability and potential theories. Among those, recurrence and transience of Markov processes have been studied by many authors under various probabilistic and analytic aspects in discrete and in continuous time. For instance, it is well-known that a d-dimensional Brownian motion consisting of d independent one-dimensional standard Brownian motions is recurrent if d = 1, 2, and transient otherwise. For more general diffusion processes, we have also many criteria for their recurrence and transience, but the criteria are not always so easy to be checked. In general, whether diffusion processes are recurrent or transient depends on their generators (see [4], [5], [6]). In this spirit, Ichihara [5] gave elegant criteria for the recurrence and transience of the diffusion process associated with a second order elliptic partial differential operator L on R d defined by where a ij (x) is a symmetric coefficient function such that the matrix A(x) := (a ij (x)) 1≤i,j≤d is strictly positive definite on R d .
Let W be the space of locally bounded Borel measurable functions on R d vanishing at the origin and let Q be a probability measure on W. In the present paper, an element of W is called an environment. Given an environment w, consider Y w = (Y w (t), P w x , x ∈ R d ), the diffusion process with generator It is well-known that Y w (t) can be constructed from the diffusion process X w (t) associated with (1.1) provided a ij = 1 2 δ ij e −w through a random time change of X w (t). We call a stochastic process Y w = (Y w (t), Q ⊗ P w x , x ∈ R d ) the diffusion process in a random environment. In the case where d = 1 and (w, Q) is a Brownian environment, Brox [1] noticed that the process Y w is a continuous version of Sinai's walk (see [15]) and showed that Y w (t) moves very slowly in some sense by the effect of the environment. Later, Brox's result was extended to a multi-dimensional diffusion process in a non-negative Lévy's Brownian environment (see [7], [11]).
Recurrence and transience of multi-dimensional diffusion processes in various random environments have been studied by many authors, in combining with Ichihara's criteria and the ergodic aspects of measure preserving transformations on the random environments. The first result on this problem was obtained by Fukushima et al. in a one-dimensional Brownian environment (see [2]). Tanaka considered the diffusion process Y w in a Lévy's Brownian environment and proved that it is to be recurrent for almost all environments in any dimension (see [18]), which made the effect of random environments on this problem quite transparent. After that, Tanaka's result was extended to a large class of multi-dimensional random environments (see [8], [10], [16], [17]). In particular, in [10], the authors considered multi-dimensional diffusion processes in multi-parameter random environments and studied their recurrence and transience. More precisely, the authors obtained some conditions for the dichotomy of recurrence and transience for d-dimensional diffusion process Y w (t) = (Y 1 w (t), Y 2 w (t), · · · , Y d w (t)) corresponding to the generator where w is a one-dimensional (semi-)stable Lévy process whose values at different d points are regarded as constituting a multi-parameter environment. In their proof, the following property of the environments was crucial: for any a 0 > 0 and θ ≥ 1 It turned out that the property (1.3) works well with Ichihara's test in studying the recurrence and transience of Y w (t). However, for instance, (1.3) does not hold if one component of w takes value identically zero. Indeed, let us consider the two-dimensional direct product of diffusion process (Y 1 w (t), B(t)) given by the pair of the Brox's diffusion Y 1 w (t) and a one-dimensional Brownian motion B(t) independent of Y 1 w (t). Let w be the environment relative to (Y 1 w (t), B(t)). Then w(x 1 , x 2 ) = w(x 1 ) and w(σ) = w(0) = 0 for σ := (0, 1) ∈ S 1 . In this sense, a (d + 1)-dimensional diffusion process in d-parameter random environments with a one-dimensional Brownian motion B(t) independent of {Y j w (t), j = 1, 2, · · · , d} is out of the framework of [10] (also of [17]).
The purpose of this paper is to study the recurrence of some multi-dimensional diffusion processes in random environments including zero potentials. For this, we introduce a new criterion for the recurrence of direct products of symmetric Markov processes motivated by Okura [12]. In the criterion, the index given by the Dirichlet forms plays an important role as representing the strength of recurrence of the associated Markov processes. The index is possible to be calculated in the case that the marginal processes are diffusion processes in self-similar random environments. As a result, we can show the recurrence of direct products of Markov processes given by the pair of a d-dimensional diffusion process in almost all environments having usual randomness, and a one-dimensional Brownian motion (see Theorems 4.3 and 4.7).
The present paper is organized as follows. In Section 2, we give criteria for the recurrence of direct products of general symmetric Markov processes including a random time changed version, and prove some lemmas on diffusion processes in non-random environments. In Section 3, we give some sufficient conditions on the random environment for the recurrence of a multi-dimensional direct product process in an ergodic random environment. In Section 4, we consider concrete examples for the result obtained in Section 3 with Gaussian and stable Lévy environments. For notational convenience, we let a ∧ b := min{a, b} for any a, b ∈ R.

Recurrence of products of Dirichlet forms
In this section, we give some analytic recurrence criteria for direct products of symmetric Dirichlet forms (or, of symmetric Markov processes). The result will be obtained by a simple obsevation for the recurrence of direct products of symmetric Markov processes due to [3] and [12].
For i = 1, 2, . . . , N , let E (i) be a locally compact separable metric space and m (i) be a positive Radon measure on E (i) with full support. Let (E (i) , F (i) ) be a symmetric regular Dirichlet form on L 2 (E (i) , m (i) ) possessing C (i) as its core. It is well-known that (E (i) , F (i) ) generates a strongly continuous Markovian semigroup (T x (i) ) be the m (i) -symmetric Hunt process associated to (E (i) , F (i) ). We say that ( Let X = (Ω, M, X(t), P x ) be the process on E defined by the product of X (i) , where We note that the marginal processes {(X (i) (t), t ≥ 0), i = 1, 2, . . . , N } are independent under P x . Let m be the product measure of {m (i) , i = 1, 2, . . . , N }. Assume that X (i) is irreducible for any i = 1, 2, . . . , N . Then, X is also to be an m-symmetric irreducible Markov process on E ([3, Proposition 3.1], [12, Theorem 2.6]). Let (E, F) be the associated Dirichlet form of X on L 2 (E, m). Then (E, F) possesses the linear span of ). Thus the Dirichlet form (E, F) is to be regular and also admits the following expressions: for u (i) ∈ F (i) (i = 1, 2, . . . , N ), Now we give some simple criteria for the non-transience of X through the marginal processes {X (i) , i = 1, 2, . . . , N } in an analytic way.
. It is then easy to see that 0 ≤ u n(k) ≤ 1 m-a.e. and u n(k) → 1 as k → ∞ m-a.e. Moreover, by (2.1) and the assumption (2.2), we have Hence (E, F) (or X) is non-transient.
For a strictly positive continuous function where τ (i) t is the right continuous inverse of the positive continuous additive functional A  [4] for the definition). Then we can obtain the following corollary as a consequence of Proposition 2.1.
for the index then the direct product process Y of {Y (i) , i = 1, 2, . . . , N } is recurrent.

Some lemmas on diffusion processes in non-random environments
Let w be a locally bounded and Borel measurable function on R d . Consider the strongly local Dirichlet form (E w , F w ) defined by where the derivatives ∂f /∂x i are taken in the sense of Schwartz distributions. Denote C ∞ 0 (R d ) by the set of all smooth functions with compact support in R d . Note that the local boundedness of w implies that C ∞ 0 (R d ) is dense in F w , in particular (E w , F w ) is regular. Let X w = (X w (t), P w x ) be the diffusion process associated with (E w , F w ). The d-dimensional Brownian motion is associated to (E 0 , F 0 ), the Dirichlet form (E w , F w ) with w ≡ 0. For r ∈ (1, ∞), let ϕ ∈ C ∞ 0 (R d ) such that 0 ≤ ϕ(x) ≤ 1 on R d , ϕ(x) = 1 on |x| ≤ 1, and ϕ(x) = 0 on |x| ≥ r. For fixed r and ϕ, define the sequence {u n } ⊂ F w by It is clear that lim n→∞ u n (x) = 1 for x ∈ R d . For notational brevity, we let for a, b ≥ 0. Then, it is easy to see by the definition of u n and the assumption on ϕ that for n ∈ N, for any ℓ ∈ N, it also follows that for n ∈ N In particular where V d denotes the volume of the unit ball in R d . On the other hand, the relation implies that for n ∈ N, Define a number n(k) ∈ N ∪ {∞} by Lemma 2.3. Let k ∈ N such that n(k) < ∞ and E w (u n(k) , u n(k) ) = 0. Then we have kr (d−2)(n(k)−1) exp −w(r n(k)−1 , r n(k) ) × 1 + (r d − 1)r dn(k) exp −w(r n(k)−1 , r n(k)+1 ) exp (−w(0, 1)) .
Proof. We note that the choice of k ∈ N and the definition of n(k) imply In view of (2.8), (2.10) and (2.11), we then have Lemma 2.4. Let k ∈ N such that n(k) < ∞ and E w (u n(k) , u n(k) ) = 0. Then we have 2 r d(d+2)/2 . Proof. In view of (2.9) for ℓ = 1 and (2.11), we see that for n ∈ N From this inequality and the definition of n(k), we have 1 k < C 2 2V d r −2n(k)+d+2 exp −w(r n(k)−1 , r n(k) ) + w(0, r n(k)−2 ) for k ∈ N such that n(k) < ∞ and E w (u n(k) , u n(k) ) = 0. Hence Applying this inequality to the upper estimate in (2.8), we can obtain the assertion.
The condition n(k) < ∞ for any k ∈ N is guaranteed in the case of d-dimensional Brownian motion (or the Dirichlet form (E 0 , F 0 )). Therefore, by virtue of Lemma 2.4, we have the following fact.
Corollary 2.5. For any k ∈ N, it holds that where C is the constant which appeared in Lemma 2.4.

Recurrence of diffusion processes in random environments
Let W be the space of locally bounded and Borel measurable functions on R d with the topology generated by the uniform convergence on compact sets. Let B(W) be the Borel σ-field of W and Q be a probability measure on (W, B(W)). We call an element w ∈ W an environment and assume that Q(w(0) = 0) = 1. For given w ∈ W, we define the Dirichlet form (E w , F w ) by (2.6) and let X w = (X w (t), P w x ) be the associated diffusion process of (E w , F w ). For r > 1 and α > 0, let T be a mapping from W to W defined by T w(x) = r −α w(rx) for x ∈ R d . We assume that Then T is a measure preserving transformation of Q. We call a space (W, B(W), Q) satisfying the condition (3.1) an α-semi-selfsimilar random environment. We say that a mapping T is weakly mixing if As in the proof of Theorem 2.2 in [8], we can prove the following lemma.
Lemma 3.1. Assume that T is weakly mixing. If A ∈ B(W) satisfies Q(A) > 0, then, for Q-almost every w ∈ W, {n ∈ N : T n w ∈ A} is an infinite set.
Let ϕ and {u n } be the functions defined as in Section 2. For given N-valued increasing sequence By using Lemma 3.1 above, we have the estimate as follows: Assume that T is a weakly mixing and Then, for Q-almost every w ∈ W, there exists an N-valued increasing sequence {n w ℓ } ℓ≥1 such that for any ε > 0 and γ ∈ [0, 1].

Proof. Set
Then we see in view of (3.1). By Lemma 3.1, there exists N ∈ B(W) such that Q(N ) = 0, and for w ∈ W \ N , w(0) = 0 and {n ∈ N : T n w ∈ A} is an infinite set. For w ∈ W \ N , let {n w ℓ : ℓ ∈ N} be a strictly increasing sequence in {n ∈ N : T n w ∈ A}. Then we have w r n w ℓ , r n w ℓ +1 − w 0, r n w ℓ < br α(n w ℓ −1) w(0, r n ℓ ) = w(0, r n ℓ +1 ).
On the other hand, by applying the first inequality in (3.3) to Lemma 2.3, it holds that for k ∈ N. From this, one can get for sufficiently large k ∈ N that where C is a constant depending on d, r, α, ϕ and a. Moreover, since we have by (2.8) and (2.11) that for k ∈ N Then, by virtue of the second and third relations in (3.3), and (3.5) hence, for γ ∈ [0, 1] and k ∈ N From this inequality, we see that the right-hand side of (3.4) is dominated by The proof is complete. Then, the (d+1)-dimensional direct product of diffusion process (X w (t), B(t)) given by the pair of the d-dimensional diffusion process X w (t) and a one-dimensional Brownian motion B(t) independent of X w (t) is recurrent for almost all environments.
Proof. We note that In view of this fact, the assumption (3.6) implies that there exists a > 0 such that for a sufficiently small ε > 0. Thus, by applying Proposition 3.2 with γ = 1/2 and b = 2a − ε, we see that Then, by Corollary 2.5 and (3.8), as k → ∞. Moreover, (3.7) implies Q(w(1, r) > a) > 0 and therefore, X w (t) is recurrent for almost all environments in view of [8,Theorem 2.2]. Hence, by virtue of Proposition 2.1, we can conclude that (X w (t), B(t)) is recurrent for almost all environments.

Applications to explicit random environments
In this section, as applications of a random environment appeared in Section 3, we consider the recurrence of the product of diffusion processes in semi-selfsimilar Gaussian and Lévy random environments.

Gaussian random environments
Let W be the space of continuous functions w on R d , with the topology generated by the uniform convergence on compact sets. We define a probability measure Q on (W, B(W)) by a Gaussian measure, that is, (w(x 1 ), w(x 2 ), . . . w(x n )) has an n-dimensional Gaussian distribution under Q, where x 1 , x 2 , . . . , x n ∈ R d for n ∈ N. We assume that Q(w(0) = 0) = 1 and E Q [w(x)] = 0 for x ∈ R d . Here E Q stands for the expectation with respect to Q. Let K be the covariance kernel of Q, that is, It is well-known that the law of a Gaussian measure is determined by the mean and the covariance kernel. First, we are going to consider a sufficient condition for (3.6) in Theorem 3.4. Then the assumption (3.6) holds.
Proof. The proof is similar to that of [8,Lemma 3.1]. So, we omit the detail and see only the sketch of the proof. By the general theory of the Gaussian system, for f ∈ L 2 (R d , dx), R d K(·, y)f (y)dy is in the Cameron-Martin space H associated to Q on L 2 (R d , dx). In particular, On the other hand, since H is dense in the support of Q, for any g ∈ H Q sup x∈R d |w(x) − g(x)| < ε > 0 for any ε > 0.
Next we consider a sufficient condition for the mixing condition. In the sequel, let r > 1, α > 0 and T be a measure preserving transformation of Q satisfying the semi-selfsimilarity (3.1) as in Section 3. We say that T is strongly mixing if It is well known that every strongly mixing transformation is weakly mixing, hence is ergodic (see [19]). Set D 1 := {x ∈ R d ; 1 < |x| < r}. Then, we see the following. For w ∈ W, let X w = (X w (t), P w x ) be the d-dimensional diffusion process associated to (E w , F w ) defined by (2.6). Note that the condition (3.6) implies Q(w(1, r) > a 0 ) > 0 for some a 0 > 0. Thus, if T on W is weakly mixing, X w is recurrent for almost all environment ([8, Theorem 2.2]). From this fact with Theorem 3.4, Lemmas 4.1 and 4.2, we have the following theorem. Then, the (d + 1)-dimensional direct product of diffusion process (X w (t), B(t)) given by the pair of X w (t) and a one-dimensional Brownian motion B(t) independent of X w (t) is recurrent for almost all environments.
Corollary 4.4. The two-dimensional direct product of diffusion process (Y w (t), B(t)) given by the pair of the Brox's diffusion process Y w (t) and a one-dimensional Brownian motion B(t) independent of Y w (t) is recurrent for almost all environments.
Proof. Let w be the two-sided Brownian motion on R under Q. In this case, w and r −1/2 w(r ·) have the same law. Furthermore, the covariance kernel K(x, y) is given by K(x, y) = (|x| ∧ |y|)1 (0,∞) (xy), x, y ∈ R.
Choose r satisfying 1 < r < 2 + √ 2. Then, since sup 1≤|x|≤r 2 |y|≤r 2 K(x, y)dy = 1 2 3) is satisfied. Moreover, it is easy to see that (4.4) is satisfied. On the other hand, we note that the Brox's diffusion process Y w (t) is a time changed process of the one-dimensional diffusion process X w (t) by the positive continuous additive functional · 0 e −w(Xw(s)) ds. Then the Dirichlet form (Ě w ,F w ) corresponding to Y w (t) is given by (2.6) replacing the underlying measure e −w(x) dx with e −2w(x) dx. It is well-known that X w (t) is recurrent (see [18]). Hence, by Corollary 2.2 and Theorem 4.3, we obtain the assertion.
By a similar argument to the proof of Proposition 2.1 in [9], we have the following. Proof. First we prove that there exists M > 0 such that for any a > 0 If α i = 2, then w i + is a Brownian motion and hence (4.5) holds. Assume that α i ∈ (0, 2). In this case, we note that the Lévy measure ν of w i + is not trivial and its Gaussian part is to be 0. Since w i + has positive jumps with a positive probability, we can choose ε ∈ (0, 1] such that ν((ε, ∞)) > 0. For i = 1, 2, . . . , d, let v i 1 , v i 2 and v i 3 be independent Lévy processes associated to Let define a random function w by w i x (i) , x = (x (1) , x (2) , . . . , x (d) ) ∈ R d .
For this w, let X w = (X w (t), P w x ) be the diffusion process associated to the Dirichlet form (E w , F w ) given by (2.6). Then, it is the d-dimensional direct products of diffusion processes in products of random environments generated by one-dimensional stable Lévy processes {(w i (x), x ∈ R), i = 1, 2, . . . , d}, that is, X w (t) = X 1 w 1 (t), X 2 w 2 (t), . . . , X d w d (t) . Now we give a sufficient condition for the recurrence of X w as follows.
Theorem 4.7. Let i = 1, 2, . . . , d. If α i = 2 or both w i + and w i − have positive jumps as positive probabilities. Then, the (d + 1)-dimensional direct product process (X w (t), B(t)) given by the pair of X w (t) and a one-dimensional Brownian motion B(t) independent of X w (t) is recurrent for almost all environments.
Proof. We remark that the components of X (w) t are independent for each environment w, because (2) , . . . , x (d) ) ∈ R d . In view of Proposition 2.1 and Lemma 4.6 we obtain the assertion.