A non-linear parabolic PDE with a distributional coefficient and its applications to stochastic analysis

TL;DR

This paper studies a non-linear parabolic PDE with a distributional coefficient in the non-linear term, establishing local and global existence and uniqueness, and applying the results to non-linear backward stochastic differential equations with distributional drivers.

Contribution

Introduces a framework for solving a non-linear PDE with distributional coefficients and applies it to stochastic analysis, specifically to backward SDEs with distributional drivers.

Findings

Proved local existence and uniqueness of solutions.

Established conditions for global existence.

Applied PDE results to stochastic differential equations.

Abstract

We consider a non-linear parabolic partial differential equation (PDE) on $\mathbb R^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity is of quadratic type in the gradient of the unknown. Under suitable conditions on the parameters we prove local existence and uniqueness of a mild solution to the PDE, and investigate properties like continuity with respect to the initial condition and blow-up times. We prove a global existence and uniqueness result assuming further properties on the non-linearity. To conclude we consider an application of the PDE to stochastic analysis, in particular to a class of non-linear backward stochastic differential equations with distributional drivers.