Self-interacting diffusions. III. Symmetric interactions

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Self-interacting diffusions. III. Symmetric interactions

Michel Benaïm and Olivier Raimond

Source: Ann. Probab. Volume 33, Number 5 (2005), 1716-1759.

Abstract

Let M be a compact Riemannian manifold. A self-interacting diffusion on M is a stochastic process solution to

$dX_{t}=dW_{t}(X_{t})-\frac{1}{t}\biggl(\int_{0}^{t}\nabla V_{X_{s}}(X_{t})\,ds\biggr)\,dt,$

where {W_t} is a Brownian vector field on M and V_x(y)=V(x,y) a smooth function. Let $\mu_{t}=\frac{1}{t}\int_{0}^{t}\delta_{X_{s}}\,ds$ denote the normalized occupation measure of X_t. We prove that, when V is symmetric, μ_t converges almost surely to the critical set of a certain nonlinear free energy functional J. Furthermore, J has generically finitely many critical points and μ_t converges almost surely toward a local minimum of J. Each local minimum has a positive probability to be selected.

First Page: Show

Primary Subjects: 60K35, 37C50

Secondary Subjects: 60H10, 62L20, 37B25

Keywords: Self-interacting random processes; reinforced processes

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1127395871
Digital Object Identifier: doi:10.1214/009117905000000251
Zentralblatt MATH identifier: 1085.60073
Mathematical Reviews number (MathSciNet): MR2165577

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