Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients

TL;DR

This paper introduces a new method for analyzing the existence and uniqueness of solutions to PDEs with polynomial growth coefficients using backward stochastic differential equations, providing a probabilistic representation of these solutions.

Contribution

It develops a novel weak convergence and compact embedding approach for BSDEs with p-growth coefficients and establishes their connection to PDE solutions.

Findings

Proves existence and uniqueness of weak solutions for PDEs with polynomial growth.

Provides a probabilistic representation of PDE solutions via BSDEs.

Introduces a new analytical framework for PDEs with unbounded coefficients.

Abstract

In this paper we develop a new weak convergence and compact embedding method to study the existence and uniqueness of the $L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of backward stochastic differential equations with p-growth coefficients. Then we establish the probabilistic representation of the weak solution of PDEs with p-growth coefficients via corresponding BSDEs.