Open Access
January 2019 Regularization by noise and flows of solutions for a stochastic heat equation
Oleg Butkovsky, Leonid Mytnik
Ann. Probab. 47(1): 165-212 (January 2019). DOI: 10.1214/18-AOP1259


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.


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Oleg Butkovsky. Leonid Mytnik. "Regularization by noise and flows of solutions for a stochastic heat equation." Ann. Probab. 47 (1) 165 - 212, January 2019.


Received: 1 November 2016; Revised: 1 February 2018; Published: January 2019
First available in Project Euclid: 13 December 2018

zbMATH: 07036336
MathSciNet: MR3909968
Digital Object Identifier: 10.1214/18-AOP1259

Primary: 35R60 , 60H15 , 60H25

Keywords: Path-by-path uniqueness , Regularization by noise , stochastic flow of solutions , Stochastic heat equation

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 1 • January 2019
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