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

2019/10/11 by Andrea Pascucci, Pascucci, Andrea, A. Pesce +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #35K70 #35R60 #60H15 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistical Mechanics and Entropy #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1910.05301

openalex publication_date 2019/10/11 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

We prove existence, regularity in H "older classes and estimates from above\nand below of the fundamental solution of the stochastic Langevin equation. This\ndegenerate SPDE satisfies the weak H "ormander condition. We use a Wentzell's\ntransform to reduce the SPDE to a PDE with random coefficients; then we apply a\nnew method, based on the parametrix technique, to construct a fundamental\nsolution. This approach avoids the use of the Duhamel's principle for the SPDE\nand the related measurability issues that appear in the stochastic framework.\nOur results are new even for the deterministic equation.\n

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