Finite Time Approach to Equilibrium in a Fractional Brownian Velocity Field |
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Authors: | Peter Horvai Tomasz Komorowski Jan Wehr |
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Affiliation: | (1) Laboratoire de Physique, ENS-Lyon and Centre de Physique Théorique, Ecole Polytechnique, Palaisseau, Lyon, France;(2) UMCS, Lublin and Institute of Mathematics, PAN, Warsaw, Poland;(3) Department of Mathematics, University of Arizona, Tucson, USA |
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Abstract: | We consider the solution of the equation r(t) = W(r(t)), r(0) = r
0 > 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 r
0, depending on whether W(r
0) 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. |
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Keywords: | Passive tracer fractional Brownian motion two point separation function |
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