We obtain existence results for the solution u of nonlocal semilinear parabolic PDEs on R d with polynomial nonlinearities in (u, ∇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.
We prove the existence of viscosity solutions for fractional semilinear elliptic PDEs on open balls with bounded exterior condition in dimension d ≥ 1. Our approach relies on a tree-based probabilistic representation based on a (2s)-stable branching processes for all s ∈ (0, 1), and our existence results hold for sufficiently small exterior conditions and nonlinearity coefficients. In comparison with existing approaches, we consider a wide class of polynomial nonlinearities without imposing upper bounds on their maximal degree or number of terms. Numerical illustrations are provided in large dimensions.
We obtain existence results for the solution u of nonlocal semilinear parabolic PDEs on R d with polynomial nonlinearities in (u, ∇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.
We present an algorithm for the numerical solution of nonlinear parabolic partial differential equations with arbitrary gradient nonlinearities. This algorithm extends the classical Feynman-Kac formula to fully nonlinear PDEs using random trees that carry information on nonlinearities on their branches. It applies to functional, nonpolynomial nonlinearities that cannot be treated by standard branching arguments and deals with gradients of any orders, avoiding the integrability issues encountered with the use of Malliavin-type weights.
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