On Probabilistic Analytical and Numerical Approaches for Divergence Form Operators with Discontinuous Coefficients

In this paper we review some recent results on stochastic analytical and numericalapproaches to parabolic and elliptic partial differential equationsinvolving a divergence form operator with a discontinuous coefficient and a compatibility transmission condition. In the one-dimensional case existence and uniqueness results for such PDEscan be obtained by stochastic methods. The probabilistic interpretation of the solutionsallows one to develop and analyze a low complexity Monte Carlo numerical resolutionmethod. In addition, it allows one to get accurate pointwise estimates for thederivatives of the solutions from which sharp convergence rate estimates are deducedfor the stochastic numerical method.A stochastic approach is also developed for the linearized Poisson-Boltzmannequation in Molecular Dynamics.As in the one-dimensional case, the probabilistic interpretation of the solution involves the solution of a SDE including a non standard localtime term related to the discontinuity interface. We presentan extended Feynman-Kac formula for the Poisson-Boltzmann equation. Thisformula justifies various probabilistic numerical methods to approximate thefree energy of a molecule and bases error analyzes.We finally present probabilistic interpretations of the non-linearizedPoisson-Boltzmann equation in terms of backward stochastic differentialequations.