Convergence of Langevin-Simulated Annealing algorithms with multiplicative noise II: Total Variation

We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for $V:{\mathbb{R}}^d\to \mathbb{R}$ a potential function to minimize, we consider the stochastic differential equation $dY_t =

• \sigma \sigma^\top \nabla V(Y_t) dt + a(t)\sigma(Y_t)dW_t + a(t)^2\Upsilon(Y_t)dt$,where$(W_t)$isaBrownianmotion,where$\sigma : $\mathbb{R}$^d \to $\mathcal{M}$_d($\mathbb{R}$)$ is an adaptive (multiplicative) noise, where $a:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a function decreasing to $0$ and where $\mathrm{{\rm Y}}$ is a correction term. Allowing $\sigma$ to depend on the position brings faster convergence in comparison with the classical Langevin equation $d{Y}_t=-\mathrm{\nabla }V(Y_t)dt+\sigma d{W}_t$. In a previous paper we established the convergence in $L^1$-Wasserstein distance of $Y_t$ and of its associated Euler scheme ${\bar{Y}}_t$ to $\text{argmin}(V)$ with the classical schedule $a(t)=A{\log }^{-1\mathrm{/}2}(t)$. In the present paper we prove the convergence in total variation distance. The total variation case appears more demanding to deal with and requires regularization lemmas.