Fastest Mixing Markov Chain on a Compact Manifold

In this paper, we address the problem of optimizing the convergence rate of a discrete-time Markov chain (DTMC), which evolves on a compact smooth connected manifold without boundary, to a specified target stationary distribution. This problem has been previously solved for a DTMC on a finite graph that converges to the uniform distribution. 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 previous work that addressed DTMCs on finite graphs, 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 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. The optimization problem admits an exact solution in the case where the manifold is a Lie group and the target measure is uniform. 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}