On stochastic Langevin and Fokker-Planck equations: The two-dimensional case

We prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution of the stochastic Langevin equation that is a degenerate SPDE satisfying the weak Hörmander condition. This problem naturally appears in stochastic filtering theory. We use a Wentzell's transform to reduce the SPDE to a PDE with random coefficients. After introducing an original notion of intrinsic solution, we apply a new method based on the parametrix technique to construct it. This approach avoids the use of the Duhamel's principle for the SPDE and the related measurability issues that appear in the stochastic framework. Our results are new even for the deterministic equation in that we prove existence and gradient estimates for the fundamental solution of equations whose coefficients are merely measurable with respect to the time variable. We also propose a different, possibly simpler proof for the Gaussian lower bound.