Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Martin boundary in non-Lipschitz domains


Authors: Richard F. Bass and Krzysztof Burdzy
Journal: Trans. Amer. Math. Soc. 337 (1993), 361-378
MSC: Primary 31C35; Secondary 60J45, 60J50
DOI: https://doi.org/10.1090/S0002-9947-1993-1100692-9
MathSciNet review: 1100692
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Martin boundary with respect to the Laplacian and with respect to uniformly elliptic operators in divergence form can be identified with the Euclidean boundary in $ {C^\gamma }$ domains, where

$\displaystyle \gamma (x) = bx\log \log (1/x)/\log \log \log (1/x),$

$ b$ small. A counterexample shows that this result is very nearly sharp.

References [Enhancements On Off] (What's this?)

  • [A] A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. 125 (1987), 495-535. MR 890161 (88k:58160)
  • [Bñ] R. Bañuelos, Intrinsic ultracontractivity and eigenvalue estimates for Schródinger operators, J. Funct. Anal. 100 (1991), 181-206. MR 1124298 (92k:35066)
  • [BBB] R. Bañuelos, R. F. Bass and K. Burdzy, Hölder domains and the boundary Harnack principle, Duke Math. J. 64 (1991), 195-200. MR 1131398 (92g:35077)
  • [BB1] R. F. Bass and K. Burdzy, A probabilistic proof of the boundary Harnack principle, Seminar on Stochastic Processes, 1989 (E. Cinlar, K. L. Chung and R. K. Getoor, eds.), Birkhäuser, Boston, Mass., 1990, pp. 1-16. MR 1042338 (92c:60106)
  • [BB2] -, A boundary Harnack principle for twisted Hölder domains, Ann. of Math. 134 (1991), 253-276.
  • [BB3] -, Lifetimes of conditioned diffusions, Probab. Theory Related Fields 91 (1992), 405-444. MR 1151804 (93e:60155)
  • [BH] R. F. Bass and P. Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains, Ann. Probab. 19 (1991), 486-508. MR 1106272 (92i:60142)
  • [Bu] K. Burdzy, Multidimensional Brownian excursions and potential theory, Longman, New York, 1987. MR 932248 (89d:60146)
  • [Do] J. L. Doob, Classical potential theory and its probabilistic counterpart, Springer, New York, 1984. MR 731258 (85k:31001)
  • [FS] E. B. Fabes and D. W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal. 96 (1986), 327-338. MR 855753 (88b:35037)
  • [HW] R. A. Hunt and R. L. Wheeden, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc. 132 (1970), 507-527. MR 0274787 (43:547)
  • [JK1] D. S. Jerison and C. E. Kenig, Boundary value problems on Lipschitz domains, Studies in Partial Differential Equations (W. Littman, ed.), Math. Assoc. Amer., Washington, D.C., 1982. MR 716504 (85f:35057)
  • [JK2] -, The Dirichlet problem in non-smooth domains, Ann. of Math. 113 (1981), 367-382.
  • [JK3] -, Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. in Math. 46 (1982), 80-147. MR 676988 (84d:31005b)
  • [M] R. S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc. 49 (1941), 137-172. MR 0003919 (2:292h)
  • [PS] S. C. Port and C. J. Stone, Brownian motion and classical potential theory, Academic Press, New York, 1978. MR 0492329 (58:11459)
  • [S] M. Sharpe, General theory of Markov processes, Academic Press, Boston, Mass., 1988. MR 958914 (89m:60169)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 31C35, 60J45, 60J50

Retrieve articles in all journals with MSC: 31C35, 60J45, 60J50


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1100692-9
Keywords: Martin boundary, Martin kernel, harmonic functions, minimal harmonic, divergence form operators, conditioned Brownian motion, $ h$-processes
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society