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 
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     title = {On unique extension of time changed reflecting brownian motions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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     mrnumber = {2548508},
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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://archive.numdam.org/articles/10.1214/08-AIHP301/

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