The Annals of Probability

Regularization by noise and flows of solutions for a stochastic heat equation

Oleg Butkovsky and Leonid Mytnik

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Motivated by the regularization by noise phenomenon for SDEs, we prove existence and uniqueness of the flow of solutions for the non-Lipschitz stochastic heat equation

\[\frac{\partial u}{\partial t}=\frac{1}{2}\frac{\partial^{2}u}{\partial z^{2}}+b\bigl(u(t,z)\bigr)+\dot{W}(t,z),\] where $\dot{W}$ is a space-time white noise on $\mathbb{R}_{+}\times\mathbb{R}$ and $b$ is a bounded measurable function on $\mathbb{R}$. As a byproduct of our proof, we also establish the so-called path-by-path uniqueness for any initial condition in a certain class on the same set of probability one. To obtain these results, we develop a new approach that extends Davie’s method (2007) to the context of stochastic partial differential equations.

Article information

Ann. Probab., Volume 47, Number 1 (2019), 165-212.

Received: November 2016
Revised: February 2018
First available in Project Euclid: 13 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 60H25: Random operators and equations [See also 47B80]

Regularization by noise stochastic heat equation path-by-path uniqueness stochastic flow of solutions


Butkovsky, Oleg; Mytnik, Leonid. Regularization by noise and flows of solutions for a stochastic heat equation. Ann. Probab. 47 (2019), no. 1, 165--212. doi:10.1214/18-AOP1259.

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Supplemental materials

  • Supplement to “Regularization by noise and flows of solutions for a stochastic heat equation”. The supplementary material provides proofs of auxiliary results related to the properties of the heat kernel.