Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On unique extension of time changed reflecting Brownian motions

Zhen-Qing Chen and Masatoshi Fukushima

Full-text: Open access

Abstract

Let D be an unbounded domain in ℝd with d≥3. We show that if D contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on ̅D is transient. Next assume that RBM X on ̅D is transient and let Y be its time change by Revuz measure 1D(x)m(x) dx for a strictly positive continuous integrable function m on ̅D. We further show that if there is some r>0 so that D̅B̅(̅0̅,̅ ̅r̅) is an unbounded uniform domain, then Y admits one and only one symmetric diffusion that genuinely extends it and admits no killings.

Résumé

Notons D un domaine non borné dans ℝd avec d≥3. Nous montrons que si D contient un domaine uniforme non borné, alors le mouvement brownien réfléchi (RBM) sur ̅D est transient. Par ailleurs, supposons que le RBM X sur ̅D est transient et notons Y son changement de temps par une mesure de Revuz 1D(x)m(x) dx pour une fonction m strictement positive, continue et intégrable sur ̅D. Nous démontrons alors que si il existe un r>0 tel que D̅B̅(̅0̅,̅ ̅r̅) soit un domaine uniformément non borné, alors Y admet une unique extension en une diffusion symétrique qui n’est jamais tuée.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 3 (2009), 864-875.

Dates
First available in Project Euclid: 4 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1249391389

Digital Object Identifier
doi:10.1214/08-AIHP301

Mathematical Reviews number (MathSciNet)
MR2548508

Zentralblatt MATH identifier
1189.60141

Subjects
Primary: 60J50: Boundary theory
Secondary: 60J60: Diffusion processes [See also 58J65]

Keywords
Reflecting Brownian motion Transience Time change Uniform domain Sobolev space BL function space Reflected Dirichlet space Harmonic function Diffusion extension

Citation

Chen, Zhen-Qing; Fukushima, Masatoshi. On unique extension of time changed reflecting Brownian motions. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 3, 864--875. doi:10.1214/08-AIHP301. https://projecteuclid.org/euclid.aihp/1249391389


Export citation

References

  • [1] M. Brelot. Étude et extensions du principe de Dirichlet. Ann. Inst. Fourier 3 (1953–1954) 371–419.
  • [2] Z.-Q. Chen. On reflected Dirichlet spaces. Probab. Theory Related Fields 94 (1992) 135–162.
  • [3] Z.-Q. Chen and M. Fukushima. One-point extensions of symmetric Markov processes by darning. Probab. Theory Related Fields 141 (2008) 61–112.
  • [4] Z.-Q. Chen, Z.-M. Ma and M. Röckner. Quasi-homeomorphisms of Dirichlet forms. Nagoya Math. J. 136 (1994) 1–15.
  • [5] J. Deny and J. L. Lions. Les espaces du type de Beppo Levi. Ann. Inst. Fourier 5 (1953–1954) 305–370.
  • [6] M. Fukushima, Y. Oshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin, 1994.
  • [7] M. Fukushima and H. Tanaka. Poisson point processes attached to symmetric diffusions. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 419–459.
  • [8] D. A. Herron and P. Koskela. Uniform, Sobolev extension and quasiconformal circle domains. J. Anal. Math. 57 (1991) 172–202.
  • [9] D. Jerison and C. Kenig. Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. Math. 46 (1982) 80–147.
  • [10] P. W. Jones. Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147 (1981) 71–78.
  • [11] Z.-M. Ma and M. Röckner. Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Berlin, 1992.
  • [12] V. G. Maz’ja. Sobolev Spaces. Springer, Berlin, 1985.
  • [13] L. Schwartz. Théorie des distributions I, II. Hermann, Paris, 1950, 1951.
  • [14] M. L. Silverstein. Symmetric Markov Processes. Lecture Notes in Math. 426. Springer, Berlin, 1974.
  • [15] J. Väisälä. Uniform domains. Tohoku Math. J. 40 (1988) 101–118.