Fatou's theorem for subordinate Brownian motions with Gaussian\n components on C^1,1 open sets

2015/12/24 by Hyunchul Park, Park, Hyunchul
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1512.07868

openalex publication_date 2015/12/24 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

We prove Fatou's theorem for nonnegative harmonic functions with respect to\nsubordinate Brownian motions with Gaussian components on bounded C^1,1 open\nsets D. We prove that nonnegative harmonic functions with respect to such\nprocesses on D converge nontangentially almost everywhere with respect to the\nsurface measure as well as the harmonic measure restricted to the boundary of\nthe domain. In order to prove this, we first prove that the harmonic measure\nrestricted to \∂ D is mutually absolutely continuous with respect to\nthe surface measure. We also show that tangential convergence fails on the unit\nball.\n

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