Ergodicity and error estimate of laws for a random splitting Langevin Monte Carlo

Abstract: The random splitting Langevin Monte Carlo could mitigate the first order bias in Langevin Monte Carlo with little extra work compared other high order schemes. We develop in this work an analysis framework for the sampling error under Wasserstein distance regarding the random splitting Langevin Monte Carlo. First, the sharp local truncation error is obtained by the relative entropy approach together with the explicit formulas for the commutator of related semi-groups. The necessary pointwise estimates of the gradient and Hessian of the logarithmic density are established by the Bernstein type approach in PDE theory. Second, the geometric ergodicity is established by accommodation of the reflection coupling. Combining the ergodicity with the local error estimate, we establish a uniform-in-time sampling error bound, showing that the invariant measure of the method approximates the true Gibbs distribution with $O(\tau^2)$ accuracy where $\tau$ is the time step. Lastly, we perform numerical experiments to validate the theoretical results.