convergence of langevin

We note that the logarithmic sobolev inequality (LSI) provides a gradient domination condition that ensures exponential convergence of langevin dynamics under the wasserstein metric\cite{otto2000generalization}. $$ \operatorname{LSI}(\pi): \quad \operatorname{KL}\left(p_t | \pi\right) \leq \frac{1}{2 \lambda} I\left(p_t | \pi\right), $$ where $I\left(p_t | \pi\right)=\int \frac{\left|\nabla p_t(x)\right|^2}{p_t(x)} d x$ is the Fisher information. This yields exponential decay: $$ \mathrm{KL}\left(p_t | \pi\right) \leq \mathrm{KL}\left(p_0 | \pi\right) \mathrm{e}^{-2 \lambda t} $$

In \cite{wibisono_sampling_2018}, the objective is to discretize this process using the Forward-Backward splitting approach while preserving convergence.