Multiplicative stochastic heat equations on the whole space

TL;DR

This paper develops a method to construct solutions for certain ill-posed multiplicative stochastic heat equations on unbounded spaces, including the parabolic Anderson model and KPZ equation, using regularity structures in weighted Besov spaces.

Contribution

It adapts the theory of regularity structures to weighted Besov spaces, enabling solutions from Dirac initial conditions for these equations on unbounded domains.

Findings

Constructed solutions for the parabolic Anderson model on A^3.

Constructed solutions for the KPZ equation on A.

Allowed initial conditions to be Dirac masses at initial time.

Abstract

We carry out the construction of some ill-posed multiplicative stochastic heat equations on unbounded domains. The two main equations our result covers are, on the one hand the parabolic Anderson model on $\mathbf{R}^3$, and on the other hand the KPZ equation on $\mathbf{R}$ via the Cole-Hopf transform. To perform these constructions, we adapt the theory of regularity structures to the setting of weighted Besov spaces. One particular feature of our construction is that it allows one to start both equations from a Dirac mass at the initial time.