Oct 15, 2013
Tuesday

11:00 AM  12:00 PM


FokkerPlanck equations, Free Energy, and Markov Processes on Graphs
Haomin Zhou (Georgia Institute of Technology)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 The classical FokkerPlanck equation is a linear parabolic equation which describes the time evolution of probability density of a stochastic process defined on an Euclidean space. Corresponding to a stochastic process, there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and entropy. In recent years, it has been shown that the FokkerPlanck equation is the gradient flow of the free energy functional defined on the Riemannian manifold of probability distributions whose inner product is generated by a 2Wasserstein distance. In this talk, we present results on similar matters for a free energy functional or Markov process defined on a graph with a finite number of vertices and edges. If $N\ge 2$ is the number of vertices of the graph, we show that the corresponding FokkerPlanck equation is a system of $N$ {\it nonlinear} ordinary differential equations defined on a Riemannian manifold of probability distributions.
However, in contrast to stochastic processes defined on Euclidean spaces, the situation is more subtle for discrete spaces. We have different choices for inner products on the space of probability distributions resulting in different FokkerPlanck equations for the same process. It is shown that there is a strong connection but also substantial discrepancies between the systems of ordinary differential equations and the classical FokkerPlanck equation on Euclidean spaces. Furthermore, both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the Riemannian manifolds of probability distributions with different metrics. Some examples will also be discussed. The work is jointly with ShuiNee Chow (Georgia Tech), Wen Huang (USTC) and Yao Li (Courant Institute).
 Supplements

Zhou
2.31 MB application/pdf


