2021/12/13 by Juraj Földes, Foldes, Juraj, David P. Herzog +1 · 1 citation
Computer Science · Economics, Econometrics and Finance · Mathematics · #34B27 #35A01 #35A02 #35G15 #35H10 #60H10 #60H30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2112.06404
openalex publication_date 2021/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study hypoelliptic stochastic differential equations (SDEs) and their connection to degenerate-elliptic boundary value problems on bounded or unbounded domains. In particular, we provide probabilistic conditions that guarantee that the formal stochastic representation of a solution is smooth on the interior of the domain and continuously approaches the prescribed boundary data at a given boundary point. The main general results are proved using fine properties of the process stopped at the boundary of the domain combined with hypoellipticity of the operators associated to the SDE. The main general results are then applied to deduce properties of the associated Green's functions and to obtain a generalization of Bony's Harnack inequality. We moreover revisit the transience and recurrence dichotomy for hypoelliptic diffusions and its relationship to invariant measures.