Motivated by the development of efficient Monte Carlo methodsfor PDE models in molecular dynamics,we establish a new probabilistic interpretation of a family of divergence formoperators with discontinuous coefficients at the interfaceof two open subsets of $\mathbb{R}^d$ . This family of operators includes the case of thelinearized Poisson-Boltzmann equation used tocompute the electrostatic free energy of a molecule.More precisely, we explicitly construct a Markov process whoseinfinitesimal generator belongs to this family, as the solution of a SDEincluding a non standard local time term related to the interfaceof discontinuity. We then prove an extendedFeynman-Kac formula for the Poisson-Boltzmann equation.This formula allows us to justifyvarious probabilistic numerical methods toapproximate the free energy of a molecule.We analyse the convergence rate of these simulation procedures andnumerically compare them on idealized molecules models.

This paper considers Schrödinger operators, and presents a probabilistic interpretation of the variation (or shape derivative) of the Dirichlet groundstate energy when the associated domain is perturbed. This interpretation relies on the distribution on the boundary of a stopped random process with Feynman-Kac weights. Practical computations require in addition the explicit approximation of the normal derivative of the groundstate on the boundary. We then propose to use this formulation in the case of the so-called fixed node approximation of Fermion groundstates, defined by the bottom eigenelements of the Schrödinger operator of a Fermionic system with Dirichlet conditions on the nodes (the set of zeros) of an initially guessed skew-symmetric function. We show that shape derivatives of the fixed node energy vanishes if and only if either (i) the distribution on the nodes of the stopped random process is symmetric; or (ii) the nodes are exactly the zeros of a skew-symmetric eigenfunction of the operator. We propose an approximation of the shape derivative of the fixed node energy that can be computed with a Monte-Carlo algorithm, which can be referred to as Nodal Monte-Carlo (NMC). The latter approximation of the shape derivative also vanishes if and only if either (i) or (ii) holds.

In this paper, we prove a Donsker theorem for one-dimensional processes generated by an operator with measurablecoefficients. We construct a random walk on any grid on the state space, using the transition probabilities of the approximated process, and the conditional average times it spends on each cell of the grid. Indeed we can compute thesequantities by solving some suitableelliptic PDE problems.

We provide a set of verifiable sufficient conditions for proving in a number of practical examples the equivalence of the martingale and the PDE approaches to the valuation of derivatives. The key idea is to use a combination of analytic and probabilistic assumptions that covers typical models in finance falling outside the range of standard results from the literature. Applications include Heston's stochastic volatility model and the Black-Karasinski term structure model.

We obtain a representation for the supercritical Dawson-Watanabe superprocessin terms of a subcritical superprocess with immigration, where the immigration at a given time is governed by the state of an underlying branching particle system. The proof requires a new result on the laws of weighted occupation times for branching particle systems.