Branching diffusion representation for nonlinear Cauchy problems and Monte Carlo approximation

Abstract

We provide probabilistic representations of the solution of some semilinear hyperbolic and high-order PDEs based on branching diffusions. These representations pave the way for an approximation of the solution by the standard Monte Carlo method, whose error estimate is controlled by the standard central limit theorem, thus partly bypassing the curse of dimensionality. We illustrate the numerical implications in the context of some popular PDEs in physics such as nonlinear Klein–Gordon equation, a simplified scalar version of the Yang–Mills equation, a fourth-order nonlinear beam equation and the Gross–Pitaevskii PDE as an example of nonlinear Schrödinger equations.