Branching diffusion representation for nonlinear Cauchyproblems and Monte Carlo approximation

TL;DR

This paper introduces a probabilistic branching diffusion method to solve high-dimensional nonlinear PDEs, enabling Monte Carlo approximation that overcomes the curse of dimensionality, with applications in physics equations.

Contribution

It presents a novel probabilistic representation for semilinear hyperbolic and high-order PDEs using branching diffusions, facilitating efficient Monte Carlo solutions.

Findings

Provides a new probabilistic representation for nonlinear PDEs.

Demonstrates Monte Carlo approximation bypassing curse of dimensionality.

Applies method to physics PDEs like Klein-Gordon and nonlinear Schrödinger.

Abstract

We provide a probabilistic representations of the solution of some semilinear hyperbolicand high-order PDEs based on branching diffusions. These representations pave theway for a Monte-Carlo approximation of the solution, thus bypassing the curse ofdimensionality. We illustrate the numerical implications in the context of some popularPDEs in physics such as nonlinear Klein-Gordon equation, a simplied scalar versionof the Yang-Mills equation, a fourth-order nonlinear beam equation and the Gross-Pitaevskii PDEas an example of nonlinear Schrodinger equations.