Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force process

Jourdain, Benjamin; Lelièvre, Tony; Roux, Raphaël

ESAIM: Modélisation mathématique et analyse numérique, Special Issue on Probabilistic methods and their applications, Tome 44 (2010) no. 5, pp. 831-865.

We study a free energy computation procedure, introduced in [Darve and Pohorille, J. Chem. Phys. 115 (2001) 9169-9183; Hénin and Chipot, J. Chem. Phys. 121 (2004) 2904-2914], 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 [Lelièvre et al., Nonlinearity 21 (2008) 1155-1181], 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 if the number of particles goes to infinity and then 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.

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