LAUSR

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 $$M\times \mathbb {R}^{n}$$M×Rn; which is obtained via a natural change of variable from a self-repelling diffusion taking the form $$\begin{aligned} dX_{t}= \sigma dB_{t}(X_t) -\int _{0}^{t}\nabla V_{X_s}(X_{t})dsdt,\qquad X_{0}=x \end{aligned}$$dXt=σdBt(Xt)-∫0t∇VXs(Xt)dsdt,X0=xwhere $$\{B_t\}$${Bt} is a Brownian vector field on M, $$\sigma >0$$σ>0 and $$V_x(y) = V(x,y)$$Vx(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 $$\mu $$μ given as the product of the normalized Riemannian measure on M and a Gaussian measure on $$\mathbb {R}^{n}$$Rn. We then prove an exponential decay to this invariant probability in $$L^{2}(\mu )$$L2(μ) and in total variation.