Analytically weak solutions to stochastic heat equations with spatially rough noise

TL;DR

This paper introduces an alternative method for establishing existence and uniqueness of solutions to one-dimensional stochastic heat equations driven by spatially rough Gaussian noise, using weak solution frameworks like variational and Krylov's $L^p$-theory.

Contribution

It provides a simpler approach to analyze solutions of stochastic heat equations with highly irregular spatial noise, improving upon previous methods.

Findings

Existence and uniqueness of solutions established using weak solution frameworks.

Simplified derivation compared to previous approaches.

Various mathematical improvements and corollaries obtained.

Abstract

In [HHL+17] the authors showed existence and uniqueness of solutions to the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and rougher than white in space (in particular, its covariance is not a measure). Here we present a simple alternative to derive such results by considering the equations in the analytically weak sense, using either the variational approach or Krylov's $L^p$-theory. Various improvements are obtained as corollaries.