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

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.