Existence and probabilistic representation of the solutions of semilinear parabolic PDEs with fractional Laplacians

TL;DR

This paper establishes the existence of solutions for nonlocal semilinear parabolic PDEs with fractional Laplacians using a novel probabilistic tree-based representation, enabling analysis of solutions and their derivatives.

Contribution

It introduces a probabilistic representation method for solutions and derivatives of nonlocal PDEs with fractional Laplacians, including numerical illustrations up to dimension 10.

Findings

Existence of solutions for fractional Laplacian PDEs demonstrated.

Probabilistic representation applicable to solutions and derivatives.

Numerical examples in high dimensions up to 10.

Abstract

We obtain existence results for the solution u of nonlocal semilinear parabolic PDEs on $\mathbb{R}^d$ with polynomial nonlinearities in $(u, \nabla u)$, using a tree-based probabilistic representation. This probabilistic representation applies to the solution of the equation itself, as well as to its partial derivatives by associating one of d marks to the initial tree branch. Partial derivatives are dealt with by integration by parts and subordination of Brownian motion. Numerical illustrations are provided in examples for the fractional Laplacian in dimension up to 10, and for the fractional Burgers equation in dimension two.