Convergence to the uniform distribution of moderately self-interacting diffusions on compact Riemannian manifolds

We consider a self-interacting diffusion $X$ on a smooth compact Riemannian manifold $\mathbb M$, described by the stochastic differential equation $$dX_t = \sqrt{2} dW_t(X_t)- \beta(t) \nabla V_t(X_t)dt,$$ where $\beta$ is suitably lower-bounded and grows at most logarithmically, and $V_t(x)=\frac{1}{t}\int_0^t V(x,X_s)ds$ for a suitable smooth function $V\colon \mathbb M^2\to\mathbb R$ that makes the term $-\nabla V_t(X_t)$ self-repelling. We prove that almost surely the normalized occupation measure $\mu_t$ of $X$ converges weakly to the uniform distribution $\mathcal U$, and we provide a polynomial rate of convergence for smooth test functions. The key to this result is showing that if $f\colon\mathbb M\to\mathbb R$ is smooth, then $\mu_{e^t}(f)$ shadows the flow generated by the ordinary differential equation $$\dot x_t=-x_t+\mathcal U(f).$$