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

Hongjie Dong, Kazuo Yamazaki

TL;DR

This work surveys the emergence and scope of convex integration in singular SPDEs driven by rough forcing, contrasting its global, nonunique solution framework with local-in-time theories like regularity structures and paracontrolled distributions. It highlights key stochastic and deterministic developments, including NS and SQG applications, and clarifies how renormalization enters via ill-defined nonlinearities. A central result, Theorem 4.1, shows uniqueness for the heat equation with odd-power damping under minimal $L^{n}$-type regularity, suggesting that convex integration is unlikely to produce nonuniqueness for the deterministic $\Phi^{4}$-type dynamics unless the forcing itself induces nonuniqueness. Overall, the paper delineates the boundaries of convex integration in SPDEs, emphasizing how noise structure (e.g., STWN) could be essential for nonuniqueness and identifying open problems in higher-dimensional and Yang–Mills–type systems.

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 $Φ^{4}$ model from quantum field theory.

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

TL;DR

This work surveys the emergence and scope of convex integration in singular SPDEs driven by rough forcing, contrasting its global, nonunique solution framework with local-in-time theories like regularity structures and paracontrolled distributions. It highlights key stochastic and deterministic developments, including NS and SQG applications, and clarifies how renormalization enters via ill-defined nonlinearities. A central result, Theorem 4.1, shows uniqueness for the heat equation with odd-power damping under minimal -type regularity, suggesting that convex integration is unlikely to produce nonuniqueness for the deterministic -type dynamics unless the forcing itself induces nonuniqueness. Overall, the paper delineates the boundaries of convex integration in SPDEs, emphasizing how noise structure (e.g., STWN) could be essential for nonuniqueness and identifying open problems in higher-dimensional and Yang–Mills–type systems.

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 model from quantum field theory.
Paper Structure (8 sections, 2 theorems, 64 equations, 1 table)

This paper contains 8 sections, 2 theorems, 64 equations, 1 table.

Table of Contents

  1. Introduction: motivation from physics and real-world applications
  2. Review on singular SPDEs and convex integration technique
  3. Review on singular SPDEs
  4. Review on the technique of convex integration: deterministic case
  5. Review on the technique of convex integration: stochastic case
  6. Convex integration technique applied to singular SPDEs
  7. Heat equation with damping of odd power
  8. Conclusion

Key Result

Theorem 4.1

Let $n \in \mathbb{N}$ be any odd number such that $n \geq 3$. Suppose that $u \in L^{n} (0,\infty; L^{n}(\mathbb{T}^{d}))$ is a weak solution to odd power according to Definition Definition 4.1. Then $u$ is unique.

Theorems & Definitions (16)

  • Remark 2.1
  • Definition 2.1
  • Remark 2.2
  • Example 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Example 3.1
  • Remark 3.1
  • Remark 3.2
  • ...and 6 more