Diffusion in random environment and the renewal theorem

TL;DR

This paper rigorously analyzes the asymptotic behavior of diffusions in Brownian environments, computing sign change distributions of an associated process and confirming predictions made by non-rigorous renormalization group methods.

Contribution

It provides a rigorous proof of the distribution of sign changes in the environment process and validates previous non-rigorous analyses using renewal theory and path decomposition.

Findings

Distribution of sign changes for the process b(x)

Probability of b maintaining the same sign over an interval

Rigorous confirmation of earlier renormalization group predictions

Abstract

According to a theorem of S. Schumacher and T. Brox, for a diffusion $X$ in a Brownian environment it holds that $(X_t-b_{\log t})/\log^2t\to 0 $ in probability, as $t\to\infty$, where $b_{\cdot}$ is a stochastic process having an explicit description and depending only on the environment. We compute the distribution of the number of sign changes for $b$ on an interval $[1,x]$ and study some of the consequences of the computation; in particular we get the probability of $b$ keeping the same sign on that interval. These results have been announced in 1999 in a non-rigorous paper by P. Le Doussal, C. Monthus, and D. Fisher and were treated with a Renormalization Group analysis. We prove that this analysis can be made rigorous using a path decomposition for the Brownian environment and renewal theory. Finally, we comment on the information these results give about the behavior of the diffusion.