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

• DOI: 10.1016/j.jde.2021.11.004
• Publication Date: 2021-11-15T22:15:56Z
• Authors: Andrea Pascucci; Antonello Pesce
• Keywords: Fokker-Planck equation; Fundamental solution stochastic; Langevin equation; Partial differential equations; Probability (math.PR); FOS: Mathematics; 60H15, 35R60, 35K70; 0101 mathematics; 01 natural sciences; Mathematics - Probability

We prove existence, regularity in H��lder classes and estimates from above and below of the fundamental solution of the stochastic Langevin equation. This degenerate SPDE satisfies the weak H��rmander condition. We use a Wentzell's transform to reduce the SPDE to a PDE with random coefficients; then we apply a new method, based on the parametrix technique, to construct a fundamental solution. 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.