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

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