Remarks on existence and uniqueness of the solution for stochastic partial differential equations

TL;DR

This paper establishes conditions for the existence and uniqueness of solutions to a broad class of stochastic partial differential equations driven by Gaussian noise, extending classical results to various operators including parabolic and hypoelliptic types.

Contribution

It provides a general framework for existence and uniqueness of SPDE solutions using Picard iteration, applicable to many operators under Dalang's condition.

Findings

Picard iteration guarantees unique solutions under specified conditions

Dalang's condition suffices for many parabolic and hypoelliptic operators

Applicable to a wide class of stochastic PDEs with Gaussian noise

Abstract

In this article we consider existence and uniqueness of the solutions to a large class of stochastic partial differential of form $\partial_t u = L_x u + b(t,u)+\sigma(t,u)\dot{W}$, driven by a Gaussian noise $\dot{W}$, white in time and spatial correlations given by a generic covariance $\gamma$. We provide natural conditions under which classical Picard iteration procedure provides a unique solution. We illustrate the applicability of our general result by providing several interesting particular choices for the operator $L_x$ under which our existence and uniqueness result hold. In particular, we show that Dalang condition given in Dalang (1999) is sufficient in the case of many parabolic and hypoelliptic operators $L_x$.