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Pathwise uniqueness by noise for singular stochastic PDEs

Davide Addona, Davide Bignamini, Carlo Orrieri, Luca Scarpa

TL;DR

Pathwise uniqueness for SPDEs with drift in differential form has been a central open problem in regularisation-by-noise. The authors develop a self-contained framework for stochastic evolution equations on separable Hilbert spaces with colored noise, identifying precise ranges of the parameters (α,β,δ,θ) for which weak uniqueness, pathwise uniqueness, and continuous dependence hold. The approach combines finite-dimensional Kolmogorov equations with an Itô–Tanaka correction and uniform contraction estimates, enabling passage to the infinite-dimensional limit and handling genuinely singular perturbations. The results are illustrated across heat-type, Burgers, Navier–Stokes, Cahn–Hilliard, and reaction–diffusion models, including dimensions up to 3 and various noise colors, thereby advancing the regularisation-by-noise theory and its practical applicability.

Abstract

Pathwise uniqueness for stochastic PDEs with drift in differential form is a main open problem in the recent literature on regularisation by noise. This paper establishes a self-contained theory in the framework of stochastic evolution equations on separable Hilbert spaces and provides a first result to address such an issue. The singularity of the drift allows to achieve novel uniqueness results for several classes of examples, ranging from fluid-dynamics to phase-separation models.

Pathwise uniqueness by noise for singular stochastic PDEs

TL;DR

Pathwise uniqueness for SPDEs with drift in differential form has been a central open problem in regularisation-by-noise. The authors develop a self-contained framework for stochastic evolution equations on separable Hilbert spaces with colored noise, identifying precise ranges of the parameters (α,β,δ,θ) for which weak uniqueness, pathwise uniqueness, and continuous dependence hold. The approach combines finite-dimensional Kolmogorov equations with an Itô–Tanaka correction and uniform contraction estimates, enabling passage to the infinite-dimensional limit and handling genuinely singular perturbations. The results are illustrated across heat-type, Burgers, Navier–Stokes, Cahn–Hilliard, and reaction–diffusion models, including dimensions up to 3 and various noise colors, thereby advancing the regularisation-by-noise theory and its practical applicability.

Abstract

Pathwise uniqueness for stochastic PDEs with drift in differential form is a main open problem in the recent literature on regularisation by noise. This paper establishes a self-contained theory in the framework of stochastic evolution equations on separable Hilbert spaces and provides a first result to address such an issue. The singularity of the drift allows to achieve novel uniqueness results for several classes of examples, ranging from fluid-dynamics to phase-separation models.

Paper Structure

This paper contains 27 sections, 11 theorems, 242 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Recent literature
  3. Discussion on the main results
  4. Main novelties
  5. Applications
  6. Technical strategy
  7. Structure of the paper
  8. Assumptions and main results
  9. Notation
  10. Assumptions
  11. Main results
  12. The Kolmogorov equation
  13. The Ornstein-Uhlenbeck semigroup
  14. The elliptic Kolmogorov equation
  15. Uniform estimates
  16. ...and 12 more sections

Key Result

Theorem 1

Assume H0--H4. Then, weak uniqueness holds for SDE with initial data in $D(A^\alpha)$, in the sense of Definition uniqueness. Moreover, if $B$ is bounded, then weak uniqueness holds for SDE with initial data in $H$, in the sense of Definition uniqueness.

Theorems & Definitions (33)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 1: Weak uniqueness
  • Theorem 2: Pathwise uniqueness
  • Theorem 3: Continuous dependence
  • Remark 2.5
  • Remark 2.6
  • Proposition 3.1
  • ...and 23 more