Self-repelling diffusions on a Riemannian manifold

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 $$dX_{t}= \sigma dB_{t}(X_t) -\int_{0}^{t}\nabla V_{X_s}(X_{t})dsdt,\qquad X_{0}=x$$ 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.