Self-repelling diffusions on a Riemannian manifold

Abstract

Let M be a compact connected oriented Riemannian manifold. The purpose of this paper is to investigate the long time behavior of a degenerate stochastic differential equation on the state space $M\times {\mathbb{R}}^n$; which is obtained via a natural change of variable from a self-repelling diffusion taking the form $$d{X}_t=\sigma d{B}_t(X_t)-{\int }_0^t\nabla V_{X_s}(X_t)dsdt,X_0=x$$ where $\{B_t\}$ is a Brownian vector field on $M$, $\sigma >0$ and $V_x(y)=V(x,y)$ is a diagonal Mercer kernel. We prove that the induced semi-group enjoys the strong Feller property and has a unique invariant probability $\mu$ given as the product of the normalized Riemannian measure on M and a Gaussian measure on ${\mathbb{R}}^n$. We then prove an exponential decay to this invariant probability in $L^2(\mu )$ and in total variation.

12 figures, 41 pages. Version 3. Typos corrected from Version 2. The presentation of Section 5 has been improved and the new introduction is more detailled than in the 1st version. Accepted for publication in Probability Theory and Related Fields

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