Random walks in space-time random media in all spatial dimensions: the full subcritical fluctuation regime

Abstract

In arbitrary spatial dimension $d\ge 1$, we study a generalized model of random walks in a time-varying random environment (RWRE) defined by a stochastic flow of kernels. We consider the quenched probability distribution of the random walker under a scaling where the time is of order $N$ and the spatial window is of size $N^{1/2}$. This spatial window may not necessarily be centered close to the origin. We show that as $N\to \infty$ there are Gaussian fluctuations up to a certain specific spatial centering radius $ψ_N$ in the tail of the quenched probability distribution, which we call the critical scale. This critical scale depends on the spatial dimension of the underlying random walk, specifically $ψ_N = O(N^{3/4})$ when $d=1$, $ψ_N = O( N/\sqrt{\log N})$ when $d=2$, and $ψ_N = O(N)$ when $d\ge 3$. In the particular case of centering the fluctuation window at the origin, our results recover and generalize some known fluctuation results for related models. However, farther from the origin, the previous literature is more sparse. The noise coefficient in the limiting Gaussian field is nontrivial and depends on the invariant measure of the two-point motion of the underlying RWRE model. We furthermore reconcile some of these coefficient formulas with previous works. As part of the proof, we introduce a general class of Markov chains with short-range interactions that admit nice estimates and limit formulas. One of the key technical results for such Markov chains is that in $d\ge 2$, one can propagate test functions backwards in time to obtain precise limiting moment formulas.