Weak nonmild solutions to some SPDEs

TL;DR

This paper introduces a new framework for establishing the existence and uniqueness of weak solutions to nonlinear stochastic heat equations driven by space-time white noise with measure-valued initial data, using a generalized stochastic convolution approach.

Contribution

It develops a novel method to define and analyze weak solutions for SPDEs with measure initial data, extending beyond traditional mild solutions.

Findings

Existence and uniqueness of weak solutions established.

Generalized stochastic convolution constructed via Young-type inequalities.

Applicable to SPDEs with measure-valued initial data.

Abstract

We study the nonlinear stochastic heat equation driven by space-time white noise in the case that the initial datum $u_0$ is a (possibly signed) measure. In this case, one cannot obtain a mild random-field solution in the usual sense. We prove instead that it is possible to establish the existence and uniqueness of a weak solution with values in a suitable function space. Our approach is based on a construction of a generalized definition of a stochastic convolution via Young-type inequalities.