The scaling of the mobility of two-dimensional Langevin dynamics in a periodic potential as the friction vanishes is not well understood for non-separable potentials. Theoretical results are lacking, and numerical calculation of the mobility in the underdamped regime is challenging. In the first part of this talk, we propose a new variance reduction method based on control variates for efficiently estimating the mobility of Langevin-type dynamics, and we present numerical experiments illustrating the performance of the approach.

In the second part of this talk, we study an importance sampling approach for calculating averages with respect to multimodal probability distributions. Traditional Markov chain Monte Carlo methods to this end, which are based on time averages along a realization of a Markov process ergodic with respect to the target probability distribution, are usually plagued by a large variance due to the metastability of the process. The estimator we study is based on an ergodic average along a realization of an overdamped Langevin process for a modified potential. We obtain an explicit expression for the optimal perturbation potential in dimension 1 and propose a general numerical approach for approximating the optimal potential in the multi-dimensional setting.