## Abstract

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 ∂u/∂t = 1/2 ∂ ^{2} u/∂z ^{2} +b (u(t, z)) + W˙(t,z), where W˙ is a space-time white noise on ℝ _{+} × ℝ and b is a bounded measurable function on 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.

Original language | English |
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Pages (from-to) | 165-212 |

Number of pages | 48 |

Journal | Annals of Probability |

Volume | 47 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2019 |

## Keywords

- Path-by-path uniqueness
- Regularization by noise
- Stochastic flow of solutions
- Stochastic heat equation

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty