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=12.2u.z2+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.