The problem of conditioning on the occupation field was investigated for the Brownian motion in 1998 independently by Aldous (Electron. Commun. Probab. 3 (1998) 79-90) and Warren and Yor (In S & eacute;minaire de Probabilit & eacute;s, XXXII (1998) 328-342, Springer) and recently for the loop soup at intensity 1/2 by Werner (In S & eacute;minaire de Probabilit & eacute;s XLVIII (2016) 481-503 Springer), Sabot and Tarr & egrave;s (Probab. Theory Related Fields 165 (2016) 559-580), and Lupu, Sabot and Tarr & egrave;s (Electron. J. Probab. 24 (2019) 1-28). We consider this problem in the case of the Brownian loop soup on the real line, and show that it is connected with a flow version of Jacobi processes, called Jacobi flow. We show that such flows naturally appear in the setting of the perturbed reflecting Brownian motion. The Jacobi flow is also related to Fleming-Viot processes, as established by Bertoin and Le Gall (Probab. Theory Related Fields 126 (2003) 261-288, Ann. Inst. Henri Poincar & eacute; Probab. Stat. 41 (2005) 307-333) and Dawson and Li (Ann. Probab. 40 (2012) 813-857). This relation allows us to interpret Perkins' disintegration theorem between Feller continuous state branching-processes and Fleming-Viot processes as a decomposition of Gaussian measures. Our approach gives a unified framework for the problems of disintegrating on the real line. The connection with Bass-Burdzy flows which was drawn in Warren (Probab. Theory Related Fields 133 (2005) 559-572) and Lupu, Sabot and Tarr & egrave;s (Electron. J. Probab. 26 (2021) 1-25) is shown to be valid in the general case.

Publication:

ANNALS OF PROBABILITY

http://dx.doi.org/10.1214/24-AOP1737

Author:

Aidekon, Elie (corresponding author)

Fudan Univ, SMS, Shanghai, Peoples R China

Email address: aidekon@fudan.edu.cn

Hu, Yueyun

Univ Sorbonne Paris Nord, LAGA, Villetaneuse, France

Email address: yueyun@math.univ-paris13.fr

Shi, Zhan

Chinese Acad Sci, AMSS, Beijing, Peoples R China

Email address:shizhan@amss.ac.cn