Existence and probabilistic representation of the solutions of semilinear parabolic PDEs with fractional Laplacians. (English) Zbl 1495.35104
Summary: 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.
MSC:
| 35K58 | Semilinear parabolic equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35R11 | Fractional partial differential equations |
| 47G30 | Pseudodifferential operators |
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 35S10 | Initial value problems for PDEs with pseudodifferential operators |
| 60J85 | Applications of branching processes |
| 65R20 | Numerical methods for integral equations |
| 60G51 | Processes with independent increments; Lévy processes |
| 60G52 | Stable stochastic processes |
| 65C05 | Monte Carlo methods |
| 45D05 | Volterra integral equations |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
Keywords:
semilinear PDEs; nonlocal PDEs; branching processes; Lévy processes; stable processes; subordination; Volterra integral equations; Monte-Carlo methodReferences:
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