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

Houssam Alrachid (1) , Tony Lelièvre (1, 2) , Raafat Talhouk (3)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MATHERIALS - MATHematics for MatERIALS
3 LU / ULB - الجامعة اللبنانية [بيروت] = Lebanese University [Beirut] = Université libanaise [Beyrouth]

We 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.

Mots clés: Fokker-Planck; Nonlocal nonlinearity; Adaptive biasing force

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Houssam Alrachid, Tony Lelièvre, Raafat Talhouk. Local and global solution for a nonlocal Fokker–Planck equation related to the adaptive biasing force process. Journal of Differential Equations, 2016, 260 (9), pp.7032 - 7058. ⟨10.1016/j.jde.2016.01.020⟩. ⟨hal-01244976⟩