Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics

¹ TOSCA project-team, INRIA Sophia Antipolis – Méditerranée, France. Mireille.Bossy@sophia.inria.fr; Nicolas.Champagnat@sophia.inria.fr; Denis.Talay@sophia.inria.fr
² IMATH, Université du sud Toulon-Var, France. maire@univ-tln.fr

Received: 31 July 2009
Revised: 23 February 2010

Abstract

Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of . This family of operators includes the case of the linearized Poisson-Boltzmann equation used to compute the electrostatic free energy of a molecule. More precisely, we explicitly construct a Markov process whose infinitesimal generator belongs to this family, as the solution of a SDE including a non standard local time term related to the interface of discontinuity. We then prove an extended Feynman-Kac formula for the Poisson-Boltzmann equation. This formula allows us to justify various probabilistic numerical methods to approximate the free energy of a molecule. We analyse the convergence rate of these simulation procedures and numerically compare them on idealized molecules models.