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September 2005 Self-interacting diffusions. III. Symmetric interactions
Michel Benaïm, Olivier Raimond
Ann. Probab. 33(5): 1716-1759 (September 2005). DOI: 10.1214/009117905000000251

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 {Wt} is a Brownian vector field on M and Vx(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 Xt. We prove that, when V is symmetric, μt converges almost surely to the critical set of a certain nonlinear free energy functional J. Furthermore, J has generically finitely many critical points and μt converges almost surely toward a local minimum of J. Each local minimum has a positive probability to be selected.

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Michel Benaïm. Olivier Raimond. "Self-interacting diffusions. III. Symmetric interactions." Ann. Probab. 33 (5) 1716 - 1759, September 2005. https://doi.org/10.1214/009117905000000251

Information

Published: September 2005
First available in Project Euclid: 22 September 2005

zbMATH: 1085.60073
MathSciNet: MR2165577
Digital Object Identifier: 10.1214/009117905000000251

Subjects:
Primary: 37C50 , 60K35
Secondary: 37B25 , 60H10 , 62L20

Keywords: Reinforced processes , Self-interacting random processes

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 5 • September 2005
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