Recurrence and transience properties of multi-dimensional diffusion processes in selfsimilar and semi-selfsimilar random environments *

We consider d -dimensional diffusion processes in multi-parameter random environments which are given by values at different d points of one-dimensional α -stable or ( r, α ) -semi-stable Lévy processes. From the model, we derive some conditions of random environments that imply the dichotomy of recurrence and transience for the d -dimensional diffusion processes. The limiting behavior is quite different from that of a d -dimensional standard Brownian motion. We also consider the direct product of a one-dimensional diffusion process in a reﬂected non-positive Brownian environment and a one-dimensional standard Brownian motion. For the two-dimensional diffusion process, we show the transience property for almost all reﬂected Brownian environments.

For a fixed W , we consider a d-dimensional diffusion process starting at 0, X W = {X k W (t), t ≥ 0, k = 1, . . . , d} whose generator is We regard values of W at different d points as constituting a multi-parameter environment. X W is constructed by d independent standard Brownian motions with a scale transformation and a time change induced by W (c.f. [9]). Each component of X W is symbolically described by where B k (t) is a one-dimensional standard Brownian motion independent of the environment (W, Q).
In the case where d = 1 and (W, Q) is a Brownian environment, Brox showed that the distribution of (log t) −2 X W (t) converges weakly as t → ∞ in [1]. This shows that X W moves very slowly by the effect of the environment. This diffusion process is a continuous model of random walks in random environments studied by Solomon [14] and Sinai [13], and X W is often called a Brox-type diffusion. Following Brox's result, Tanaka studied the cases of α-stable Lévy environments and showed the convergence theorem with the scaling (log t) −α X W (t) under the assumption that Q{W (1) > 0} > 0 in [19]. Tanaka's results were extended to the cases of (r, α)-semi-stable Lévy environments in [16].
In view of the subdiffusive property of the Brox-type diffusion, we expect to see an exotic limiting behavior for multi-dimensional Brox-type diffusions. We now give a brief review of investigations related to multi-dimensional Brox-type diffusions. Fukushima et al. showed the recurrence of the diffusion process whose generator is where |x| = x 2 1 + x 2 2 + x 2 3 + · · · + x 2 d and W is a one-dimensional standard Brownian motion in [6]. In the case where the environment is Lévy's Brownian motion W (x) with a multi-dimensional time, Tanaka showed the recurrence of the diffusion process for almost all environments in any dimension in [20]. These results were shown by Ichihara's recurrent test introduced in [8]. Mathieu studied asymptotic behavior of multi-dimensional diffusion processes in random environments by using the Dirichlet form and showed the convergence theorem for the case where the environment is a nonnegative reflected Lévy's Brownian motion in [11]. Following this study, Kim obtained several limit theorems of the multi-dimensional diffusion processes in [10]. He also ECP 22 (2017), paper 4.
showed the convergence theorem in the case where the random environment consists of d independent one-dimensional reflected non-negative Brownian motions, which is a model studied in [17].
Recently, Gantert et al. showed the recurrence of d independent random walks in random environments satisfying a uniform ellipticity condition for the environments. They used estimates of quenched return probabilities to the origin of the one-dimensional random walks in random environments in [7]. In [18], the multi-dimensional diffusion process consisting of d independent Brox-type diffusions was studied and the recurrence of this process for almost all environments in any dimension was shown. The multidimensional diffusion process is a continuous time analogue of the random walks in [7]. We remark that the recurrence property of the multi-dimensional diffusion process is derived without the uniform ellipticity condition for the environments.
Following the previous studies, we consider limiting behavior of diffusion processes in (r, α)-semi-stable Lévy environments to be given by (1.1) and (1.2), which are extensions of the models studied in [7] and [18]. We call an increasing (r, α)-semi-stable or α-stable Lévy process a subordinator. We obtain some conditions of the random environments which imply the dichotomy of recurrence and transience of d-dimensional diffusion processes corresponding to the generator (1.3) as follows.
Q} is a subordinator, then X W is transient for almost all environments in any dimension.
Next, we consider d-dimensional diffusion processes consisting of d independent Brox-type diffusions. Let Q k be the probability measure on W such that (iii) the Lévy processes on positive and negative sides are independent.
We define an environment (W, Q) by {(W k , Q k ), k = 1, . . . , d} with each (W k , Q k ) being independent. We remark that Suzuki studied the case where d = 1 with independent an α-stable and a β-stable Lévy environments, and obtained some convergence theorems in [15].
For a fixed W, we consider a d-dimensional diffusion process starting at 0, On the d-dimensional diffusion processes, we obtain the following dichotomy theorem.
through a simple calculation. Since semi-stable or stable Lévy processes are bounded on finite time intervals almost surely (cf. Chapter VIII in [3]), the Dirichlet form W (x j ) dx and its closure (denoted by E again) is a local regular Dirichlet form (see Section 3 in [5]). Furthermore, E is either transient or recurrent, and the property is the same as that of the diffusion process corresponding to the generator (2.1). Hence, we study limiting behavior of the process Y W corresponding to the generator The following proposition about a property of (r, α)-semi-stable Lévy environments is crucial for showing the recurrence of Y W .
where S d−1 is the surface of the d-dimensional unit ball centered at the origin.
Proof. If {W (x)} is a trivial strictly semi-stable or stable Lévy process, then the assumption that Q{W (1) > 0} > 0 implies that there exists a positive constant c such that W (x) = cx almost surely (cf. Remark 13.17 in [12]). Hence, we obtain (2.3) for any d.
We next consider non-trivial cases.
is the α-stable Lévy environment, then r = 1. This and the property of the support of the α-stable Lévy process (cf. Theorem 24.10 in [12]) imply the assertion. If W (x) is the (r, α)-semi-stable environment, then we also have that Q{W (1) > a 0 + 1} > 0.
Using the local boundedness of Lévy processes, for any u > 0 and any M > 0 we find Hence, we obtain (2.4).
In the case where d ≥ 2, we set φ(x) = (a 0 + 2d)x 2 − 2. As d j=1 σ 2 j = 1, we find that  In the same manner as the case where d = 1, we consider the positive side and shall show that For W (x), we define the first exit time from the interval [−1, φ(r) + 1] by τ . Let 0 < α < 2. Then, the distribution of W (1) (denoted by µ) is strictly (r, α)semi-stable or α-stable. Let ν(dx) be the Lévy measure of µ. We can write that W = W 1 + W 2 + W 3 , where they are independent Lévy processes whose Lévy measures are restrictions of ν such that ν 1 := [ν] {x≥φ(r)+2} , ν 2 := [ν] {x≤−1} and ν 3 := [ν] {−1<x<φ(r)+2} , respectively. For k = 1 and 2, we set the first jump time of W k by τ k and set the total mass of ν k by J k . We remark that the assumption implies that J 1 > 0. Then, we obtain (2.10) Using the local boundedness of Lévy processes, we find that the probability that W 3 (x) does not exit from [−1, φ(r) + 1] with jumps induced by ν 3 until the time 1/ √ a 0 + 2d is positive. This positivity of the probability, (2.9) and (2.10) imply that   Proof of (i) of Theorem 1.1. In the case where d = 1, the scale function of the generator (2.2) is given by x 0 exp{W (s)}ds. The assumption that Q{W (1) > 0} > 0 implies that lim sup |s|→∞ W (s) = ∞ almost surely. Hence, we obtain assertion (i) from a general theory of one-dimensional diffusion processes.
In the case where d ≥ 2, we use Ichihara's recurrent test ( [8] and [5]). It is sufficient to show that for almost all environments (2.14) where dσ is the normalized uniform measure on S d−1 . First, we consider the case of the (r, α)-semi-stable Lévy environment. For each s ≥ 1, we can find n ∈ Z and θ ∈ [1, r) such that s = r n θ, which implies that the left-hand side of (2.14) is equal to ∞ n=0 r n(2−d) We set a measure preserving transformation for (W, Q) such that for n = 0, 1, 2, . . . T n W (s) := r −n/α W (r n s), s ≥ 1.
Then, the family of transformations {T n } is ergodic (for detail, see [4]). Setting (2.17) where C d and C d,r denote the surface area of the unit sphere and a positive constant determined by d and r, respectively. We choose m 0 which satisfies that for any n n(2 − d) log r + r n/α m 0 ≥ 0.
on a subset of S d−1 with positive surface measure.
We next consider non-trivial cases. Since {−W (x)} is a strictly semi-stable Lévy process, Theorem 14.7 in [12] implies that γ 0 ≡ 0. For the environment, we obtain that (the right-hand side of (2.21)) For the α-stable Lévy environment, we use semi-selfsimilarity of the environment and can show the assertion in the same way.
In the case where d ≥ 2, for a fixed W the components of X W are independent since X W is constructed by d independent standard Brownian motion. Hence we obtain the transience in any dimension.

Proof of Theorem 1.2
In the same reason as mentioned in the proof of Theorem 1.1, we study limiting behavior of the process Y W (t) corresponding to the generator Proof of (i) of Theorem 1.2. We set ECP 22 (2017), paper 4.
We also set R := R 1 × · · · × R d \ {(0, . . . , 0)} and S d−1 := S d−1 ∩ R . On R , a point x and a pair s > 0 and σ ∈ S d−1 have one-to-one correspondence to each other such that x j = s γj σ j . We thus denote a function of x by s = s(x) which satisfies that d j=1 x j s γj 2 = 1.
We set By arguments of Ichihara's recurrent test in [8] and [5], if J = ∞ for any and almost all environments, then we obtain the assertion. Settingγ := min{α j , β j , j = 1, . . . , d}, we have that where C d,γ denotes a positive constant determined by d andγ. We set a measure preserving transformation for (W j , Q j ) such that for all t ∈ R T (j) t W j (σ j ) := e −t W j (e tγj σ j ), σ ∈ S d−1 .
A direct product of strong mixing transformations is strong mixing (cf. For each component of W, the positive side and the negative side are independent. In addition, W j 's are also independent. These and (2.8) imply that for any d ≥ 2 Hence, we obtain that for any , any a 0 > 0 and any t ∈ R Q {M (t) > a 0 } > 0. (2.23) We choose m 0 > 0 which satisfies that for any t ≥ 0 t{2γ − (γ 1 + · · · + γ d )} + e t m 0 ≥ 0. Then, we obtain that (the right-hand side of (2.22)) Proof of (ii) of Theorem 1.

Transience of a two-dimensional diffusion process
In [7], Gantert et al. considered a two-dimensional random walk {X n , Y n , n ∈ N} such that {X n } is a random walk in the random environment studied by Sinai [13], {Y n } is a centered random walk that converges weakly to a strictly α-stable Lévy process with α ∈ (1, 2] under a suitable scaling, and they are independent. They showed the recurrence of the random walk for almost all environments. In [19], Tanaka studied a one-dimensional diffusion process in a reflected non-positive Brownian environment (−|W |, Q) and showed the convergence theorem with the same scaling as that of the diffusion process studied by Brox [1]. Denoting Tanaka's diffusion process by {X −|W | (t)}, we consider a direct product of {X −|W | (t)} and a one-dimensional standard Brownian motion (denoting by {B 2 (t)}). For the two-dimensional diffusion process, we obtain the transience property. Proof. By applying Ichihara's transient test in [8]     Hence, we obtain that for almost all environments and a fixed ∈ (0, 1)