Localization and number of visited valleys for a transient diffusion in random environment

Pierre Andreoletti,Alexis Devulder

doi:10.1214/ejp.v20-3173

Abstract

We consider a transient diffusion in a $(-\kappa/2)$-drifted Brownian potential $W_{\kappa}$ with $0 < \kappa < 1$. We prove its localization at time $t$ in the neighborhood of some random points depending only on the environment, which are the positive $h_t$-minima of the environment, for $h_t$ a bit smaller than $\log t$.We also prove an Aging phenomenon for the diffusion, a renewal theorem for the hitting time of the farthest visited valley, and provide a central limit theorem for the number of valleys visited up to time $t$. The proof relies on adecomposition of the trajectory of $W_{\kappa}$ in the neighborhood of$h_t$-minima, with the help of results of A. Faggionato, and on a precise analysis of exponential functionals of $W_{\kappa}$ and of $W_{\kappa}$ Doob-conditioned to stay positive.;

Highlights

These diffusions in random potentials are considered as continuous time analogues of random walks in random environment (RWRE)
We focus on a quenched study, which has attracted much interest for transient RWRE in the last few years, see for example the works of N
We prove some intermediate results, which we think will be useful for obtaining new results about the maximum local time of X, as explained later in this introduction

Summary

Introduction and notation

≈: (I1− + I2−)(I1+ + I2+)e1, τ2(ht) eWκ(x)−Wκ(m 2)dx e1 τ2(ht/2) τ2(ht) which is the product of 5 independent random variables, the first 4 depending only on the potential Wκ, and e1 being independent of Wκ This approximation, the asymptotics of the Laplace transforms of I1−, I2−, I1+ and I2+ provided by Lemma 4.2, and some technical calculations help us to prove (1.7) as claimed in our Proposition 4.1. We turn to the proof of the localization, that is, our Theorem 1.3 To this aim, using the previous renewal results, we prove that with probability nearly 1, at time t, the diffusion X has already spent a quite large amount of time in the last valley visited, that is, on [L−Nt, LNt], and that X(t) still belongs to this interval. Some events are denoted by Eij.k for some i, j, k; for example E34.7 is the event number 3 introduced in the proof of Lemma 4.7

Standard valleys and path decomposition of the potential

Quasi-Independence in the trajectories of X

Time spent in a standard valley

Proof of the main results

Conclusion

Proofs of some technical estimates related to the environment