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

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.

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.

DOI : https://doi.org/10.1214/08-AIHP301
Classification:  60J50,  60J60
Keywords: 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},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {3},
     year = {2009},
     pages = {864-875},
     doi = {10.1214/08-AIHP301},
     zbl = {1189.60141},
     mrnumber = {2548508},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 45 (2009) no. 3, pp. 864-875. doi : 10.1214/08-AIHP301. http://www.numdam.org/item/AIHPB_2009__45_3_864_0/

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