A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees

Abstract

ABSTRACT. The solutions to a large class of semi-linear parabolic PDEs are given in terms of expectations of suitable functionals of a tree of branching particles. A sufficient, and in some cases necessary, condition is given for the integrability of the stochastic representation, using a companion scalar PDE.In cases where the representation fails to be integrable a sequence of pruned trees is constructed, producing a approximate stochastic representations that in some cases converge, globally in time, to the solution of the original PDE.

Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees

“…Such truncation can be seen as ''a pruning" of the full random tree, which amounts to keep only few trees possessing a certain number of branches. Such a procedure has been used in literature before, but mainly to expand the set of initial-conditions for which a probabilistic representation can be found (see [13], e.g.). Performing such a pruning, the additional truncation error…”

Stochastic PDEs and Lack of Regularity

“…Proof Explicit calculation using Fourier series or abstract H δ bounds on products of H α and H β in the space H −γ [99]. For d = 2 see [18, Lemma A.1] for d = 1 see [19].…”

“…In the following, we need local existence and uniqueness of solutions only for a simplified system, where it will be obvious. But let us remark that either the results in [19] for a probabilistic representation via branching processes yields the existence of solutions for the system ( 21), or we can modify our results in Section 3 and use existence of solutions for the original PDE.…”

“…We do not give a detailed proof. One idea of proof is to use the probabilistic representation of solutions from [19]. Another approach would be to show that the standard fixedpoint-iteration for the mild formulation (see (23) below) preserves the spaces of functions with values in Σ and Σ + .…”

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

“…The branching mechanism has also been applied in McKean [25] to the KPP equation, and to the blow-up of solutions of Fujita [11] equations of the form ∂u(t, x)/∂t = u(t, x) + cu β (t, x) in López-Mimbela [24], see also Chakraborty and López-Mimbela [9] for the existence of solutions of parabolic PDEs with power series nonlinearities. Related arguments have also been applied to Fourier-transformed PDEs in order to treat the Navier-Stokes equation by the use of stochastic cascades in Le Jan and Sznitman [23], see also Blömker et al [7] for the representation of Fourier modes for the solution of class of semilinear parabolic PDEs.…”