Speaker:  David P. Herzog, Associate Professor, Department of Mathematics, Iowa State University

Title: Ergodicity and convergence to equilibrium for Langevin dynamics with general potentials.

Abstract:

Langevin dynamics is Newton's law for the motion of N particles subject to friction, thermal fluctuations and potential forces. Aside from its relevance in statistical mechanics, its discretizations are used in Markov chain Monte Carlo to draw samples from its explicit, and moldable, stationary distribution by running the system long enough. Because of its ballistic, as opposed to diffusive, behavior, it is believed to have a better rate of convergence to equilibrium when compared to stochastic gradient dynamics (also known as ``overdamped Langevin"). However, the precise mechanisms leading to geometric ergodicity of the system are more nuanced than stochastic gradient dynamics, especially because the SDE is degenerate elliptic and damping only explicitly acts on the momentum directions. This has led to an abundance of research on the topic. The goal of this talk is to give an overview of methods used to establish convergence to equilibrium for Langevin dynamics forced by a wide class of potential functions. In the process, we will give an overview of results.  Particular attention will be paid to both probabilistic and functional analytic methods.