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

Andrea Pascucci; Antonello Pesce

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arXiv · 1910.05301

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

Andrea Pascucci·Antonello Pesce

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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.

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BibTeX RIS

Andrea Pascucci, Antonello Pesce. 2019-10-11. On stochastic Langevin and Fokker-Planck equations: the two-dimensional case. https://arxiv.org/abs/1910.05301

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