Fastest Mixing Markov Chain on a Compact Manifold

Abstract

In this paper, we address the problem of optimizing the convergence rate of a discrete-time Markov chain, which evolves on a compact smooth connected manifold without boundary, to a specified target stationary distribution. This problem has been previously solved for a discrete-time Markov chain on a finite graph that converges to the uniform distribution. In contrast to this previous work, we consider arbitrary positive target measures that are supported on the entire state space of the system and are absolutely continuous with respect to the Riemannian volume. Similar to the earlier work, we pose the optimization problem in terms of maximizing the spectral gap of the operator that pushes forward measures, also known as the forward operator. Prior to formulating the optimization problem, we prove the existence of a Kolmogorov forward operator that can stabilize the class of measures that we consider. In addition, we prove the existence of an optimal solution to our problem. Lastly, we develop a numerical scheme for solving the optimization problem and validate our approach on a simulated system that evolves on a torus in $mathbb{R}^3$.