  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 initial
conditions. Here, $L$ and $A$ are certain partial differential operators
involving multiplication by smooth functions and are of the order two and one
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 $(L, A)$ 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.
