Reversible stochastic flows associated with nonlinear SPDEs

In this paper we study the reversibility of a stochastic flow based on its integral representation when the diffusion vector field commutes with drift vector fields. This allows us to write the complete solution of stochastic characteristic system as a composition of the deterministic flows generated by the drift and diffusion vector fields combined with the Wiener process acting in the equation. This explicit integral representation of the solution implied reversibility of the complete solution taking into account the reversibility of each deterministic flow in the finite composition. The unique solution satisfies a nonlinear SPDE.