2024/02/18 by Lucia Caramellino, Caramellino, Lucia, Cristian Mendico +1
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #Statistical Mechanics and Entropy #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2402.11524
openalex publication_date 2024/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We address the well-posedness of subelliptic Fokker-Planck equations arising from stochastic control problems, as well as the properties of the associated diffusion processes. Here, the main difficulty arises from the possible polynomial growth of the coefficients, which is related to the growth of the family of vector fields generating the first layer of the associated Lie algebra. We prove the existence and uniqueness of the energy solution, and its representation as the transition density of the underlying subelliptic diffusion process. Moreover, we show its Holder continuity in time w.r.t.the Fortet-Mourier distance, where the Holder seminorm depends on the degree of homogeneity of the vector fields. Finally, we provide a probabilistic proof of the Feyman-Kac formula, as a consequence of the uniform boundedness in finite time intervals of all moments.