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 $$d{X}_t=\sqrt{2}d{W}_t(X_t)-\beta (t)\mathrm{\nabla }{V}_t(X_t)dt,$$ 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 $V\text{\hspace{0.17em}⁣}:{\mathbb{M}}^2\to \mathbb{R}$ that makes the term $-\mathrm{\nabla }{V}_t(X_t)$ 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 $f\text{\hspace{0.17em}⁣}:\mathbb{M}\to \mathbb{R}$ is smooth, then ${\mu }_{e^t}(f)$ shadows the flow generated by the ordinary differential equation $${\dot{x}}_t=-x_t+\mathcal{U}(f)\mathrm{.}$$