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

TL;DR

This paper establishes existence, regularity, and estimates for the fundamental solution of a degenerate stochastic Langevin equation satisfying the weak H"ormander condition, using a novel parametrix approach that simplifies the analysis.

Contribution

It introduces a new method based on the parametrix technique to construct fundamental solutions for degenerate SPDEs, avoiding traditional measurability issues.

Findings

Proved existence and regularity of the fundamental solution.

Derived upper and lower estimates for the solution.

Applied the method to both stochastic and deterministic equations.

Abstract

We prove existence, regularity in H\"older classes and estimates from above and below of the fundamental solution of the stochastic Langevin equation. This degenerate SPDE satisfies the weak H\"ormander 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.