Illinois Journal of Mathematics

Fatou’s theorem for subordinate Brownian motions with Gaussian components on $C^{1,1}$ open sets

Hyunchul Park

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Abstract

We prove Fatou’s theorem for nonnegative harmonic functions with respect to killed subordinate Brownian motions with Gaussian components on bounded $C^{1,1}$ open sets $D$. We prove that nonnegative harmonic functions with respect to such processes on $D$ converge nontangentially almost everywhere with respect to the surface measure as well as the harmonic measure restricted to the boundary of the domain. In order to prove this, we first prove that the harmonic measure restricted to $\partial D$ is mutually absolutely continuous with respect to the surface measure. We also show that tangential convergence fails on the unit ball.

Article information

Source
Illinois J. Math., Volume 60, Number 3-4 (2016), 761-790.

Dates
Received: 15 September 2016
Revised: 3 April 2017
First available in Project Euclid: 22 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1506067290

Digital Object Identifier
doi:10.1215/ijm/1506067290

Mathematical Reviews number (MathSciNet)
MR3705444

Zentralblatt MATH identifier
1376.31003

Subjects
Primary: 31B25: Boundary behavior 60J75: Jump processes
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J50: Boundary theory

Citation

Park, Hyunchul. Fatou’s theorem for subordinate Brownian motions with Gaussian components on $C^{1,1}$ open sets. Illinois J. Math. 60 (2016), no. 3-4, 761--790. doi:10.1215/ijm/1506067290. https://projecteuclid.org/euclid.ijm/1506067290


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References

  • H. Aikawa, Boundary Harnack principle and Martin boundary for a uniform domain, J. Math. Soc. Japan 53 (2001), no. 1, 119–145.
  • H. Aikawa, T. Kilpeläinen, N. Shanmugalingam and X. Zhong, Boundary Harnack principle for $p$-harmonic functions in smooth Euclidean domains, Potential Anal. 26 (2007), no. 3, 281–301.
  • R. F. Bass, Probabilistic techniques in analysis, Probability and Its Applications (New York), Springer-Verlag, New York, 1995.
  • R. F. Bass and D. You, A Fatou theorem for $\alpha$-harmonic functions, Bull. Sci. Math. 127 (2003), no. 7, 635–648.
  • R. F. Bass and D. You, A Fatou theorem for $\alpha$-harmonic functions in Lipschitz domains, Probab. Theory Related Fields 133 (2005), no. 3, 391–408.
  • A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions. II, Math. Proc. Cambridge Philos. Soc. 42 (1946), 1–10.
  • K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song and Z. Vondraček, Potential analysis of stable processes and its extensions, Lecture Notes in Mathematics, vol. 1980, Springer-Verlag, Berlin, 2009. Edited by Piotr Graczyk and Andrzej Stos.
  • K. Bogdan and B. Dyda, Relative Fatou theorem for harmonic functions of rotation invariant stable processes in smooth domains, Studia Math. 157 (2003), no. 1, 83–96.
  • Z.-Q. Chen, R. Durrett and G. Ma, Holomorphic diffusions and boundary behavior of harmonic functions, Ann. Probab. 25 (1997), no. 3, 1103–1134.
  • Z.-Q. Chen, P. Kim and R. Song, Heat kernel estimates for $\Delta+\Delta^{\alpha/2}$ in $C^{1,1}$ open sets, J. Lond. Math. Soc. (2) 84 (2011), no. 1, 58–80.
  • Z.-Q. Chen, P. Kim, R. Song and Z. Vondraček, Sharp Green function estimates for $\Delta+\Delta ^{\alpha/2}$ in $C^{1,1}$ open sets and their applications, Illinois J. Math. 54 (2010), no. 3, 981–1024.
  • Z.-Q. Chen, P. Kim, R. Song and Z. Vondraček, Boundary Harnack principle for $\Delta+\Delta^{\alpha/2}$, Trans. Amer. Math. Soc. 364 (2012), no. 8, 4169–4205.
  • Z.-Q. Chen and R. Song, Martin boundary and integral representation for harmonic functions of symmetric stable processes, J. Funct. Anal. 159 (1998), no. 1, 267–294.
  • K. L. Chung and Z. X. Zhao, From Brownian motion to Schrödinger's equation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 312, Springer-Verlag, Berlin, 1995.
  • P. Fatou, Séries trigonométriques et séries de Taylor, Acta Math. 30 (1906), no. 1, 335–400.
  • G. B. Folland, Real analysis: Modern techniques and their applications, 2nd ed., Pure and Applied Mathematics (New York), Wiley, New York, 1999. A Wiley-Interscience Publication.
  • P. Kim, Fatou's theorem for censored stable processes, Stochastic Process. Appl. 108 (2003), no. 1, 63–92.
  • P. Kim, Relative Fatou's theorem for $(-\Delta)^{\alpha /2}$-harmonic functions in bounded $\kappa$-fat open sets, J. Funct. Anal. 234 (2006), no. 1, 70–105.
  • P. Kim, R. Song and Z. Vondraček, Potential theory of subordinate Brownian motions revisited, Stochastic analysis and applications to finance, Interdiscip. Math. Sci., vol. 13, World Scientific, Hackensack, NJ, 2012, pp. 243–290.
  • P. Kim, R. Song and Z. Vondraček, Potential theory of subordinate Brownian motions with Gaussian components, Stochastic Process. Appl. 123 (2013), no. 3, 764–795.
  • H. Kunita and T. Watanabe, Markov processes and Martin boundaries. I, Illinois J. Math. 9 (1965), 485–526.
  • J. E. Littlewood, Mathematical notes (4): On a theorem of Fatou, J. Lond. Math. Soc. (2) S1–2 (1927), no. 3, 172–176.
  • K. Michalik and M. Ryznar, Relative Fatou theorem for $\alpha$-harmonic functions in Lipschitz domains, Illinois J. Math. 48 (2004), no. 3, 977–998.
  • P. W. Millar, First passage distributions of processes with independent increments, Ann. Probab. 3 (1975), 215–233.
  • K. Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original. Revised by the author.
  • H. Šikić, R. Song and Z. Vondraček, Potential theory of geometric stable processes, Probab. Theory Related Fields 135 (2006), no. 4, 547–575.