Probabilistic representations of solutions of nonlinear PDEs

Abstract

We provide new probabilistic representations for solutions of nonlinear differential equations through the use of branching processes. These stochastic methods are used to derive local existence criteria and can be implemented for Monte Carlo simulations of the solutions. The first part of the thesis is devoted to parabolic and elliptic PDEs involving pseudo-differential operators such as the fractional Laplacian and polynomial nonlinearities involving the gradient of the solution. In the second part, we focus on representations for ODEs and parabolic PDEs involving smooth general nonlinearity of the derivatives of any order by the use of a new stochastic structure named coding trees. These methods require strong integrability conditions to ensure the expectations are finite. We also present new methods to derive criteria for the blow-up of some nonlocal problems.