Existence, uniqueness and convergence of a particle approximation for the Adaptive Biasing Force process

We study a free energy computation procedure, introduced in [4, 7], which relies on the long-time behavior of a nonlinear stochastic differential equation. This nonlinearity comes from a conditional expectation computed with respect to one coordinate of the solution. The long-time convergence of the solutions to this equation has been proved in [10], under some existence and regularity assumptions. In this paper, we prove existence and uniqueness under suitable conditions for the nonlinear equation, and we study a particle approximation technique based on a Nadaraya-Watson estimator of the conditional expectation. The particle system converges to the solution of the nonlinear equation as the number of particles goes to infinity and the kernel used in the Nadaraya-Watson approximation tends to a Dirac mass. We derive a rate for this convergence, and illustrate it by numerical examples on a toy model.