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

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.