Journal of the Mathematical Society of Japan

Reflections at infinity of time changed RBMs on a domain with Liouville branches

Zhen-Qing CHEN and Masatoshi FUKUSHIMA

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $Z$ be the transient reflecting Brownian motion on the closure of an unbounded domain $D\subset \mathbb{R}^d$ with $N$ number of Liouville branches. We consider a diffuion $X$ on $\overline{D}$ having finite lifetime obtained from $Z$ by a time change. We show that $X$ admits only a finite number of possible symmetric conservative diffusion extensions $Y$ beyond its lifetime characterized by possible partitions of the collection of $N$ ends and we identify the family of the extended Dirichlet spaces of all $Y$ (which are independent of time change used) as subspaces of the space $\mathrm{BL}(D)$ spanned by the extended Sobolev space $H_e^1(D)$ and the approaching probabilities of $Z$ to the ends of Liouville branches.

Article information

Source
J. Math. Soc. Japan, Volume 70, Number 2 (2018), 833-852.

Dates
First available in Project Euclid: 18 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1524038675

Digital Object Identifier
doi:10.2969/jmsj/07027645

Mathematical Reviews number (MathSciNet)
MR3787741

Zentralblatt MATH identifier
06902443

Subjects
Primary: 60J50: Boundary theory
Secondary: 60J65: Brownian motion [See also 58J65] 31C25: Dirichlet spaces

Keywords
transient reflecting Brownian motion time change Liouville domain Beppo Levi space approaching probability quasi-homeomorphism zero flux

Citation

CHEN, Zhen-Qing; FUKUSHIMA, Masatoshi. Reflections at infinity of time changed RBMs on a domain with Liouville branches. J. Math. Soc. Japan 70 (2018), no. 2, 833--852. doi:10.2969/jmsj/07027645. https://projecteuclid.org/euclid.jmsj/1524038675


Export citation

References

  • N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, De Gruyter, 1991.
  • M. Brelot, Etude et extension du principe de Dirichlet, Ann. Inst. Fourier, 5 (1953/54), 371–419.
  • Z.-Q. Chen, On reflecting diffusion processes and Skorokhod decompositions, Probab. Theory Relat. Fields, 94 (1993), 281–315.
  • Z.-Q. Chen and M. Fukushima, On unique extension of time changed reflecting Brownian motions, Ann. Inst. Henri Poincaré Probab. Statist., 45 (2009), 864–875.
  • Z.-Q. Chen and M. Fukushima, Symmetric Markov Processes, Time Change and Boundary Theory, Princeton University Press, 2011.
  • Z.-Q. Chen and M. Fukushima, One-point reflections, Stochastic Process Appl., 125 (2015), 1368–1393.
  • Z.-Q. Chen, Z.-M. Ma and M. Röckner, Quasi-homeomorphisms of Dirichlet forms, Nagoya Math. J., 136 (1994), 1–15.
  • J. Deny and J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier, 5 (1953/54), 305–370.
  • X. Fang, M. Fukushima and J. Ying, On regular Dirichlet subspaces of $H^1(I)$ and associated linear diffusions, Osaka J. Math., 42 (2005), 27–41.
  • M. Fukushima, On boundary conditions for multi-dimensional Brownian motions with symmetric resolvent densities, J. Math. Soc. Japan, 21 (1969), 58–93.
  • M. Fukushima, From one dimensional diffusions to symmetric Markov processes, Stochastic Process Appl., 120 (2010), 590–604.
  • M. Fukushima, On general boundary conditions for one-dimensional diffusions with symmetry, J. Math. Soc. Japan, 66 (2014), 289–316.
  • M. Fukushima, Liouville property of harmonic functions of finite energy for Dirichlet forms, to appear in Stochastic Partial Differential Equations and Related Fields, Springer Proc. in Math. and Stat.
  • M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Grruyter, 1994, 2nd Edition, 2010.
  • M. Fukushima and M. Takeda, Markov Processes (in Japanese), Baifukan, Tokyo, 2008, Chinese translation by P. He, (ed. J. Ying), Science Press, Beijing, 2011.
  • M. Fukushima and M. Tomisaki, Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps, Probab. Theory Relat. Fields, 106 (1996), 521–557.
  • A. Grigor'yan and L. Saloff-Coste, Heat kernels on manifolds with ends, Ann. Inst. Fourier, Grenoble 59 (2009), 1917–1997.
  • D. A. Herron and P. Koskella, Uniform, Sobolev extension and quasiconformal circle domains, J. Anal. Math., 57 (1991), 172–202.
  • P. W. Jones, Quasiconformal mappings and extendibility of functions in Sobolev spaces, Acta Math., 147 (1981), 71–88.
  • Y. Kuz'menko and S. Molchanov, Counterexamples to Liouville-type theorems, Moscow Univ. Math. Bull., 34 (1979), 35–39.
  • Z.-M. Ma and M. Röckner, Introduction to the Theory of (non-symmetric) Dirichlet Forms, Springer, 1992.
  • V. G. Maz'ja, Sobolev Spaces, Springer, 1985.
  • R. G. Pinsky, Transience/recurrence for normally reflected Brownian motion in unbounded domains, Ann. Probab., 37 (2009), 676–686.