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 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.Mathematics Subject Classification. 35Q60, 92C40, 60J60, 65C05, 65C20, 68U20.

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We describe a Monte Carlo method for the numerical computation of the principal eigenvalue of the Laplace operator in a bounded domain with Dirichlet conditions. It is based on the estimation of the speed of absorption of the Brownian motion by the boundary of the domain. Various tools of statistical estimation and different simulation schemes are developed to optimize the method. Numerical examples are studied to check the accuracy and the robustness of our approach.

Using a sequential control variates algorithm, we compute Monte Carlo approximations of solutions of linear partial differential equations connected to linear Markov processes by the Feynman-Kac formula. It includes diffusion processes with or without absorbing/reflecting boundary and jump processes. We prove that the bias and the variance decrease geometrically with the number of steps of our algorithm. Numerical examples show the efficiency of the method on elliptic and parabolic problems. Introduction.We are concerned with the numerical evaluation of E(Ψ(X s : s ≥ t)|X t = x), where (X t ) t is a Markov process (with linear dynamics) and where Ψ belongs to a class of functionals related to Feynman-Kac representations. These issues arise, for example, in physics in the computations of the solution of diffusion equations (see [CDL + 89]), or in finance in the pricing of European options (see [DG95] and the references therein). Monte Carlo methods are usually used to evaluate these expectations for high-dimensional problems or when the functionals are complex. They give a rather poor approximation because of a slow convergence as σ/ √ M , M being the number of simulations and σ 2 the relative variance. A better accuracy can nevertheless be reached by using relevant variance-reduction tools like, for instance, the control variates method or importance sampling [Hal70, New94]. One of the most performing tools is the sequential Monte Carlo approach which consists in using iteratively these variance-reduction ideas [Hal62, Hal70, Boo89]. Using, respectively, importance sampling and control variates, this approach has been recently developed in [BCP00] for Markov chains and in [Mai03] for the numerical integration of multivariate smooth functions. We have introduced in [GM04] a sequential Monte Carlo method to solve the Poisson equation with Dirichlet boundary conditions over square domains. This method was based on Feynman-Kac computations of pointwise solutions combined with a global approximation on Tchebychef polynomials [BM97]. Pointwise solutions were computed using walk-on-spheres (WOS) simulations of stopped Brownian motion, which induces a simulation error due to the absorption layer thickness. We have nevertheless observed a geometric reduction up to a limit of both the simulation error and the variance with the number of steps of the algorithm. The global error was comparable to standard deterministic spectral methods [BM97] while avoiding the resolution of a linear system. Our goal here is twofold:• to extend the scope of the approach to general Markov processes connected to linear elliptic and parabolic Dirichlet problems;

We give a stochastic representation of the principal eigenvalue of some homogeneous neutron transport operators. Our construction is based upon the Feynman-Kac formula for integral transport equations, and uses probabilistic techniques only. We develop a Monte Carlo method for criticality computations. We numerically test this method on various homogeneous and inhomogeneous problems, and compare our results with those obtained by standard methods.

We introduce Monte Carlo methods to compute the solution of elliptic equations with pure Neumann boundary conditions. We first prove that the solution obtained by the stochastic representation has a zero mean value with respect to the invariant measure of the stochastic process associated to the equation. Pointwise approximations are computed by means of standard and new simulation schemes especially devised for local time approximation on the boundary of the domain. Global approximations are computed thanks to a stochastic spectral formulation taking into account the property of zero mean value of the solution. This stochastic formulation is asymptotically perfect in terms of conditioning. Numerical examples are given on the Laplace operator on a square domain with both pure Neumann and mixed Dirichlet-Neumann boundary conditions. A more general convection-diffusion equation is also numerically studied.

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