On the equivalence of the static and dynamic point of view for diffusions in a random environment

Tom Schmitz

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Submission date: 25. Sep. 2008 (revised version: January 2009)
Pages: 24
published in: Stochastic processes and their applications, 119 (2009) 8, p. 2501-2522 
DOI number (of the published article): 10.1016/j.spa.2008.12.008
Bibtex
MSC-Numbers: 60K37, 82D30
Keywords and phrases: diffusion in random environment, environment viewed from the particle, invariant measures, almost linear coordinates
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Abstract:
We study the equivalence of the static and dynamic point of view for diffusions in a random environment in dimension one. First we prove that the static and dynamic distributions are equivalent if and only if either the speed in the law of large numbers does not vanish, or b/a is a.s. the gradient of a stationary function, where a and b are the covariance coefficient resp. the local drift attached to the diffusion. We moreover show that the equivalence of the static and dynamic point of view is characterized by the existence of so-called ``almost linear coordinates''.