On stochastic Langevin and Fokker-Planck equations: the two-dimensional case
Probability
2019-10-14 v1
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.
Keywords
Cite
@article{arxiv.1910.05301,
title = {On stochastic Langevin and Fokker-Planck equations: the two-dimensional case},
author = {Andrea Pascucci and Antonello Pesce},
journal= {arXiv preprint arXiv:1910.05301},
year = {2019}
}