On unique extension of time changed reflecting brownian motions
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 864-875.

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 D∖̅B̅(̅0̅,̅ ̅r̅) soit un domaine uniformément non borné, alors Y admet une unique extension en une diffusion symétrique qui n'est jamais tuée.

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 D∖̅B̅(̅0̅,̅ ̅r̅) is an unbounded uniform domain, then Y admits one and only one symmetric diffusion that genuinely extends it and admits no killings.

DOI : 10.1214/08-AIHP301
Classification : 60J50, 60J60
Mots-clés : reflecting brownian motion, transience, time change, uniform domain, Sobolev space, BL function space, reflected Dirichlet space, harmonic function, diffusion extension 
@article{AIHPB_2009__45_3_864_0,
     author = {Chen, Zhen-Qing and Fukushima, Masatoshi},
     title = {On unique extension of time changed reflecting brownian motions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {864--875},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {3},
     year = {2009},
     doi = {10.1214/08-AIHP301},
     mrnumber = {2548508},
     zbl = {1189.60141},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/08-AIHP301/}
}
TY  - JOUR
AU  - Chen, Zhen-Qing
AU  - Fukushima, Masatoshi
TI  - On unique extension of time changed reflecting brownian motions
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
SP  - 864
EP  - 875
VL  - 45
IS  - 3
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/08-AIHP301/
DO  - 10.1214/08-AIHP301
LA  - en
ID  - AIHPB_2009__45_3_864_0
ER  - 
%0 Journal Article
%A Chen, Zhen-Qing
%A Fukushima, Masatoshi
%T On unique extension of time changed reflecting brownian motions
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 864-875
%V 45
%N 3
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/08-AIHP301/
%R 10.1214/08-AIHP301
%G en
%F AIHPB_2009__45_3_864_0
Chen, Zhen-Qing; Fukushima, Masatoshi. On unique extension of time changed reflecting brownian motions. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 864-875. doi : 10.1214/08-AIHP301. http://www.numdam.org/articles/10.1214/08-AIHP301/

[1] M. Brelot. Étude et extensions du principe de Dirichlet. Ann. Inst. Fourier 3 (1953-1954) 371-419. | Numdam | MR | Zbl

[2] Z.-Q. Chen. On reflected Dirichlet spaces. Probab. Theory Related Fields 94 (1992) 135-162. | MR | Zbl

[3] Z.-Q. Chen and M. Fukushima. One-point extensions of symmetric Markov processes by darning. Probab. Theory Related Fields 141 (2008) 61-112. | MR | Zbl

[4] Z.-Q. Chen, Z.-M. Ma and M. Röckner. Quasi-homeomorphisms of Dirichlet forms. Nagoya Math. J. 136 (1994) 1-15. | MR | Zbl

[5] J. Deny and J. L. Lions. Les espaces du type de Beppo Levi. Ann. Inst. Fourier 5 (1953-1954) 305-370. | Numdam | MR | Zbl

[6] M. Fukushima, Y. Oshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin, 1994. | MR | Zbl

[7] M. Fukushima and H. Tanaka. Poisson point processes attached to symmetric diffusions. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 419-459. | Numdam | MR | Zbl

[8] D. A. Herron and P. Koskela. Uniform, Sobolev extension and quasiconformal circle domains. J. Anal. Math. 57 (1991) 172-202. | MR | Zbl

[9] D. Jerison and C. Kenig. Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. Math. 46 (1982) 80-147. | MR | Zbl

[10] P. W. Jones. Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147 (1981) 71-78. | MR | Zbl

[11] Z.-M. Ma and M. Röckner. Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Berlin, 1992. | Zbl

[12] V. G. Maz'Ja. Sobolev Spaces. Springer, Berlin, 1985. | MR

[13] L. Schwartz. Théorie des distributions I, II. Hermann, Paris, 1950, 1951. | MR | Zbl

[14] M. L. Silverstein. Symmetric Markov Processes. Lecture Notes in Math. 426. Springer, Berlin, 1974. | MR | Zbl

[15] J. Väisälä. Uniform domains. Tohoku Math. J. 40 (1988) 101-118. | MR | Zbl

Cité par Sources :