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

Mireille Bossy (1) , Nicolas Champagnat (1) , Sylvain Maire (2) , Denis Talay (1)

1 TOSCA - Simuler et calibrer des modèles stochastiques
2 MNC - Modélisation Numérique et Couplages

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 $\mathbb{R}^d$. 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.

Mots clés: Divergence form operator; Poisson-Boltzmann equation; Feynman-Kac formula; random walk on sphere algorithm

Domaines: Probabilités [math.PR]

Mireille Bossy, Nicolas Champagnat, Sylvain Maire, Denis Talay. Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics. ESAIM: Mathematical Modelling and Numerical Analysis, 2010, 44 (5), pp.997-1048. ⟨10.1051/m2an/2010050⟩. ⟨inria-00459411⟩