An existence and uniqueness theory to stochastic partial differential equations involving pseudo-differential operators driven by space-time white noise

TL;DR

This paper develops a new weak formulation for stochastic PDEs with pseudo-differential operators, allowing for rough symbols and irregular data, ensuring well-posedness under minimal regularity conditions.

Contribution

It introduces a relaxed framework for stochastic PDEs with pseudo-differential operators, accommodating rough symbols and irregular data while maintaining well-posedness.

Findings

Allows symbols to be random and irregular

Removes regularity and ellipticity conditions

Includes operators with no regularity gain or integrability improvement

Abstract

In this paper, we aim to develop a new weak formulation that ensures well-posedness for a broad range of stochastic partial differential equations with pseudo-differential operators whose symbols depend only on time and spatial frequencies. The main focus of this paper is to relax the conditions on the symbols of pseudo-differential operators and data while still ensuring that the stochastic partial differential equations remain well-posed in a weak sense. Specifically, we allow symbols to be random and remove all regularity and ellipticity conditions on them. As a result, our main operators include many interesting rough operators that cannot generate any regularity gain or integrability improvement from the equations. In addition, our data do not need to be regular or possess finite stochastic moments.