RECURRENCE OF MULTI-DIMENSIONAL DIFFUSION PROCESSES IN BROWNIAN ENVIRONMENTS

. We consider limiting behavior of multi-dimensional diﬀusion processes in two types of Brownian environments. One is given values at diﬀerent d points of a one- dimensional Brownian motion, which is supposed to be a multi-parameter environment, and other is given by d independent one-dimensional Brownian motions. We show recur- rence of multi-dimensional diﬀusion processes in both Brownian environments above for any dimension and almost all environments. Their limiting behavior is quite diﬀerent from that of ordinary multi-dimensional Brownian motion. We also consider cases of reﬂected Brownian environments.


Introduction and results.
It is well-known that an R d -dimensional Brownian motion, which is formed by d independent one-dimensional Brownian motions, is recurrent if d = 1 or 2, and transient otherwise. We consider such a problem for multi-dimensional diffusion processes in Brownian environments.
Let W be the space of R-valued continuous functions with vanishing at 0 and let Q be the Wiener measure on W. We call (W, Q) a Brownian environment. For a given W (x), we consider a multi-dimensional diffusion process in Brownian environment (W, Q) X W = {X k W (t), k = 1, 2, 3, ..., d} whose generator is d k=1 We treat a one-dimensional Brownian motion and regard its values at different d points as a multi-parameter environment. Such an X W is constructed from d independent Brownian motions by a scale change and a time change (c.f. [5]). Each component of X W above is symbolically described by where B k (t) is a Brownian motion independent of a Brownian environment W .
In the case where d = 1, the diffusion X W is a continuous model of random walks in a random environment by Sinai([9]), and recurrent for almost all Brownian environments (see also [10]). Brox studied the case and showed that the distribution of (log t) −2 X W (t) converges weakly as t → ∞ in [1], that is, X W moves very slowly by the effect of environments.
Because of the subdiffusive property of X W , we expect to see exotic limiting behavior of a multi-dimensional diffusion process X W . In this paper, we consider this problem for Brownian environments. Our main theorem is as follows: For almost all Brownian environments, X W corresponding to the generator (1) is recurrent for any dimension d = 1, 2, 3, . . ..
Some investigations related to multi-dimensional diffusion processes in random environments have been conducted. In [2], Fukushima, Nakao and Takeda obtained recurrent property of the diffusion process with a generator where |x| = x 2 1 + x 2 2 + · · · + x 2 d and W is a one-dimensional Brownian environment. Their environment is a function of the distance from the origin. In the case where an environment is a Lévy's Brownian motion with a multi-dimensional time, Tanaka showed that the diffusion process is recurrent for almost all environments in [13]. For a non-negative reflected Lévy's Brownian environment, Mathieu obtained some results of long time behavior in [8]. Our model is different from theirs, but it gives a certain multi-dimensional extension of the previous work by Brox. In the case of d independent one-dimensional Brownian environments, we have the following: Theorem 1.2. Let W be a set of d independent copies of a Brownian environment (W, Q) and let X W be a multi-dimensional diffusion process in d independent Brownian environments W with a generator d k=1 Then for any dimension d = 1, 2, 3, . . ., X W is recurrent for almost all environments.

2.
Proof of theorems. We use Ichihara's recurrent test studied in [3] in a similar manner to previous works in [2] and [13]. We can show that the generator (1) equals to through a simple calculation. We set Using them, we construct a time changed process as whose generator is given by Since τ t is strictly increasing, we have the following lemma: Hence, we study limiting behavior of Y W . To show the assertion, we use the following lemma: .
where S and T are the correlation matrices of (W (x 1 ), W (x 2 ), . . . , W (x m )) and of (W (x 1 ), As r ij (t) converges to 0 as t → 0, the ergodicity of {T t } can be shown in the same way as the theorem in [4] or Theorem 9 in [7].
Proof. We set ϕ(s) = 2a 0 s 2 . Then For some positive a, we let W ∞ < a denote {W : sup u∈[0,1] |W (u)| < a}. Then (4) implies that Proof of Theorem 1.1. According to Ichihara's recurrent test for the generator (1), it is enough to show that for almost all environments where dσ is the normalized uniform measure on S d−1 . Setting r = e t for the left hand side of (5), we obtain that where C d denotes the surface area of the unit sphere. We set M (s) := min Let a be a positive number such that for any t > 0 We set a 0 := inf a. Then Lemma 2. = P min By this convergence and Lemma 2.3, we obtain that which concludes Theorem 1.1.
Proof of Theorem 1.2. In the same way as the case of X W , we can show that recurrent or transient property of X W with the generator (2) coincides with that of Y W whose generator is given by Ergodicity of {T t } under Q ⊗d is shown in a similar way to the case for the measure Q, namely we consider the covariance matrix of (W 1 (x 1 ), . . . , W d (x 1 ), W 1 (x 2 ), . . . , W d (x 2 ), . . . , W 1 (x m ), . . . , W d (x m ), and for any x, y ∈ R and i, j ∈ N implies the ergodicity.
We can show that Q sup for each component. Since W j 's are independent, (7) implies This positivity and the ergodic property conclude Theorem 1.2.
3. The cases of reflected Brownian environments. Tanaka studied the diffusion process in a non-negative or non-positive reflected Brownian environment in [12]. He showed that each distribution of (log t) −2 X(t) also converges weakly. Through a simple argument, they are recurrent for almost all environments when d = 1. Following the study, Kim obtained some limit theorems of multi-dimensional diffusion processes in [6]. He gave an example that the random environment is given by d independent one-dimensional reflected non-negative Brownian environments studied in [11]. In the case of reflected Brownian environments, we obtain the following results: Theorem 3.1. (i) Suppose non-negative reflected Brownian environment, namely we consider a multi-dimensional diffusion process in a non-negative reflected Brownian environment X |W | with a generator d k=1 Then, for almost all environments X |W | is recurrent for any dimension d = 1, 2, 3, . . .. (ii) A diffusion process in non-positive reflected Brownian environment X −|W | , whose generator is is recurrent for d = 1 and transient for d = 2, 3, 4, . . ., for almost all environments.
Proof. The assertion (i) is shown in the same way as those for showing the cases X W , and we omit the proof. On the assertion (ii), recurrence of the one-dimensional diffusion process is easily obtained from a general theory of one-dimensional diffusion process. Hence, we prove the case where d ≥ 2. For the diffusion process X −|W | with the generator (9), Lemma 2.1 implies that it is sufficient to consider the process Y −|W | whose generator is According to Ichihara's transient test for (10), it is enough to show that for almost all the environments Without loss of generality, we can assume that there exists a positive a such that 0 < a < σ 1 ≤ σ 2 < 1. For such a σ = (σ 1 , σ 2 ) ∈ S 1 , we take the expectation with respect to Q as follows: For any x > 0 Using the inequality ∞ ξ e −u 2 /2 du ≤ 1 ξ e −ξ 2 /2 for ξ > 0, we obtain that (the right hand side of (12)) ≤ ∞ 1 1 r 2 √ 2π e r/2 1 r 1/2 e −r/2 dr ≤ ∞ 1 r −3/2 dr = 2 < ∞, which implies assertion.