Remarks on the convex integration technique applied to singular stochastic partial differential equations

TL;DR

This paper reviews the application of convex integration to singular stochastic PDEs, highlighting its potential and limitations, especially regarding the non-uniqueness in the $\

Contribution

It provides a critical review of convex integration's role in solving singular SPDEs and discusses its limitations in proving non-uniqueness for the $\

Findings

Convex integration is promising for constructing solutions to singular SPDEs.

Proving non-uniqueness via convex integration appears unlikely for the $\

Proposes that alternative methods may be needed for certain singular SPDEs.

Abstract

Singular stochastic partial differential equations informally refer to the partial differential equations with rough random force that leads to the products in the nonlinear terms becoming ill-defined. Besides the theories of regularity structures and paracontrolled distributions, the technique of convex integration has emerged as a possible approach to construct a solution to such singular stochastic partial differential equations. We review recent developments in this area, and also demonstrate that an application of the convex integration technique to prove non-uniqueness seems unlikely for a particular singular stochastic partial differential equation, specifically the $\Phi^{4}$ model from quantum field theory.