Nonlinear Backward Stochastic Evolutionary Equations Driven by a Space-Time White Noise

TL;DR

This paper investigates the existence and uniqueness of solutions for nonlinear backward stochastic evolutionary equations driven by space-time white noise, introducing new a priori estimates and employing duality arguments.

Contribution

It provides the first well-posedness results for nonlinear backward stochastic evolutionary equations driven by space-time white noise, with novel a priori estimates and a duality approach.

Findings

Established a priori estimates for linear equations

Proved existence and uniqueness for nonlinear equations

Provided an illustrative example

Abstract

We study the well solvability of nonlinear backward stochastic evolutionary equations driven by a space-time white noise. We first establish a novel a priori estimate for solution of linear backward stochastic evolutionary equations, and then give an existence and uniqueness result for nonlinear backward stochastic evolutionary equations. A dual argument plays a crucial role in the proof of these results. Finally, an example is given to illustrate the existence and uniqueness result.