Finite Time Approach to Equilibrium in a Fractional Brownian Velocity Field

Abstract

We consider the solution of the equation r(t) = W(r(t)), r(0) = r0 > 0 where W(⋅) is a fractional Brownian motion (f.B.m.) with the Hurst exponent α∈ (0,1). We show that for almost all realizations of W(⋅) the trajectory reaches in finite time the nearest equilibrium point (i.e. zero of the f.B.m.) either to the right or to the left of r0, depending on whether W(r0) is positive or not. After reaching the equilibrium the trajectory stays in it forever. The problem is motivated by studying the separation between two particles in a Gaussian velocity field which satisfies a local self-similarity hypothesis. In contrast to the case when the forcing term is a Brownian motion (then an analogous statement is a consequence of the Markov property of the process) we show our result using as the principal tools the properties of time reversibility of the law of the f.B.m., see Lemma 2.4 below, and the small ball estimate of Molchan, Commun. Math. Phys.205 (1999) 97–111.