Existence and uniqueness of stochastic PDEs associated with the forward equations: An approach using alternate norms

In this paper, we consider a class of stochastic PDEs and associated PDEs with regular initial conditions. Here, the diffusion coefficient operator [Formula: see text] and drift coefficient operator [Formula: see text] are certain partial differential operators involving multiplication by smooth functions and are of the order one and two respectively, and in special cases are associated with finite dimensional diffusion processes. This PDE also includes Kolmogorov’s forward equation (Fokker–Planck equation) as a special case. We first prove a Monotonicity inequality for the pair [Formula: see text] and using this inequality, we obtain the existence and uniqueness of strong solutions to the Stochastic PDE and the PDE. In addition, a stochastic representation for the solution to the PDE is also established.