Journal
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
Volume 120, Issue 7, Pages 1215-1246Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/j.spa.2010.03.009
Keywords
Self-interacting diffusion; Stationary measures; Double-well potential; Perturbed dynamical system; Laplace's method; Fixed point theorem; McKean-Vlasov stochastic differential equations
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We investigate the existence of invariant measures for self-stabilizing diffusions. These stochastic processes represent roughly the behavior of some Brownian particle moving in a double-well landscape and attracted by its own law. This specific self-interaction leads to nonlinear stochastic differential equations and permits pointing out singular phenomena like non-uniqueness of associated stationary measures. The existence of several invariant measures is essentially based on the non-convex environment and requires generalized Laplace's method approximations. (C) 2010 Elsevier B.V. All rights reserved.
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