Local and global solution for a nonlocal Fokker-Planck equation related to the adaptive biaising force processes

TL;DR

This paper establishes the global existence, uniqueness, and regularity of solutions for a nonlocal nonlinear Fokker-Planck equation used in adaptive importance sampling for molecular dynamics, employing semigroup theory and supersolution estimates.

Contribution

It provides the first rigorous proof of well-posedness for this specific nonlocal Fokker-Planck equation in the context of molecular dynamics sampling.

Findings

Proved global existence and uniqueness of solutions.

Established regularity of solutions in various function spaces.

Applied semigroup theory and supersolution techniques.

Abstract

We prove global existence, uniqueness and regularity of the mild, Lp and classical solution of a non-linear Fokker-Planck equation arising in an adaptive importance sampling method for molecular dynamics calculations. The non- linear term is related to a conditional expectation, and is thus non-local. The proof uses tools from the theory of semigroups of linear operators for the local existence result, and an a priori estimate based on a supersolution for the global existence result.