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We consider stochastic PDEs \[dY_t = L(Y_t)\, dt + A(Y_t).\, dB_t, t > 0\]and associated PDEs \[du_t = L u_t\, dt, t > 0\] with regular initialconditions. Here, $L$ and $A$ are certain partial differential operatorsinvolving multiplication by smooth functions and are of the order two and onerespectively, and in special cases are associated with finite dimensionaldiffusion processes. This PDE also includes Kolmogorov's Forward Equation(Fokker-Planck Equation) as a special case. We first prove a Monotonicityinequality for the pair $(L, A)$ and using this inequality, we obtain theexistence and uniqueness of strong solutions to the Stochastic PDE and the PDE.In addition, a stochastic representation for the solution to the PDE is alsoestablished.

Existence and Uniqueness of Stochastic PDEs associated with the Forward Equations: An Approach using Alternate Norms