Communications in Mathematical Sciences

Volume 17 (2019)

Number 4

A data-driven method for the steady state of randomly perturbed dynamics

Pages: 1045 – 1059



Yao Li (Department of Mathematics and Statistics, University of Massachusetts, Amherst, Mass., U.S.A.)


We demonstrate a data-driven method to solve for the invariant probability density function of a randomly perturbed dynamical system. The key idea is to replace the boundary condition of numerical schemes by a least squares problem corresponding to a reference solution, which is generated by Monte Carlo simulation. With this method we can solve for the invariant probability density function in any local area with high accuracy, regardless of whether the attractor is covered by the numerical domain.


invariant probability density function, Monte Carlo, Fokker–Planck equation

2010 Mathematics Subject Classification

37M25, 65C05, 65C30, 65N99

This work is supported by NSF DMS-1813246.

Received 17 May 2018

Accepted 17 March 2019

Published 25 October 2019