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

We prove global existence, uniqueness and regularity of the mild, LpLp 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.