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

Hongjie Dong; Kazuo Yamazaki

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arXiv · 2601.09990

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

Hongjie Dong·Kazuo Yamazaki

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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.

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Hongjie Dong, Kazuo Yamazaki. 2026-01-15. Remarks on the convex integration technique applied to singular stochastic partial differential equations. https://arxiv.org/abs/2601.09990

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