Local and global solution for a nonlocal Fokker–Planck equation related to the adaptive biasing force process

Abstract

International audienceWe prove global existence, uniqueness and regularity of the mild, L p 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

Spherical Hellinger--Kantorovich Gradient Flows

“…Here we opted to fix E, f , m satisfying some hypotheses, but it is possible to state all the assumptions in terms of f only, and then reconstruct E and m in a relevant way, see Section 3. 1. Some examples are presented in Section 3.4.…”

“…The model (1.8)-(1.11) can be viewed as a reactive nonlinear equation of Fokker-Planck type, in the spirit of [21], with conservation of mass. Reaction-diffusion problems with conservation of mass were studied in [41,26,44,45,1,25,17], see also the references therein. On the other hand, after a change of variables, our problem fits into the framework of fitness-driven models of population dynamics, and might be applicable to some human societies.…”

“…The model (1.8)- (1.11) can be viewed as a reactive nonlinear equation of Fokker-Planck type, in the spirit of [21], with conservation of mass. Reaction-diffusion problems with conservation of mass were studied in [41,26,44,45,1,25,17], see also the references therein.…”