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 $$\frac{\partial u}{\partial t}=\frac12\frac{\partial^2 u}{\partial z^2} + b(u(t,z)) + \dot{W}(t,z), $$ 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. This extends recent results of Davie (2007) to the context of stochastic partial differential equations.