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:Rd→RV\colon\mathbb{R}^{d}\to\mathbb{R} a potential function to minimize, we consider the stochastic differential equation d⁢Yt=−σ⁢σ⊤⁢∇V⁢(Yt)⁢d⁢t+a⁢(t)⁢σ⁢(Yt)⁢d⁢Wt+a⁢(t)2⁢Υ⁢(Yt)⁢d⁢tdY_{t}=-\sigma\sigma^{\top}\nabla V(Y_{t})\,dt+a(t)\sigma(Y_{t})\,dW_{t}+a(t)^{2}\Upsilon(Y_{t})\,dt, where (Wt)(W_{t}) is a Brownian motion, σ:Rd→Md⁢(R)\sigma\colon\mathbb{R}^{d}\to\mathcal{M}_{d}(\mathbb{R}) is an adaptive (multiplicative) noise, a:R+→R+a\colon\mathbb{R}^{+}\to\mathbb{R}^{+} is a function decreasing to 0 and where Υ is a correction term. Allowing 𝜎 to depend on the position brings faster convergence in comparison with the classical Langevin equation d⁢Yt=−∇V⁢(Yt)⁢d⁢t+σ⁢d⁢WtdY_{t}=-\nabla V(Y_{t})\,dt+\sigma\,dW_{t}. In a previous paper, we established the convergence in L1L^{1}-Wasserstein distance of YtY_{t} and of its associated Euler scheme Y¯t\bar{Y}_{t} to argmin⁡(V)\operatorname{argmin}(V) with the classical schedule a⁢(t)=A⁢log−1/2⁡(t)a(t)=A\log^{-1/2}(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.