The stochastic Jacobi flow

The problem of conditioning on the occupation field was investigated for the Brownian motion in 1998 independently by Aldous [4] and Warren and Yor [34] and recently for the loop soup at intensity $1/2$ by Werner [35], Sabot and Tarr\`es [30], and Lupu, Sabot and Tarr\`es [22]. 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 give a pathwise construction of this flow simultaneously for all parameters by means of a common Brownian motion, via the perturbed reflecting Brownian motion. The Jacobi flow is related to Fleming--Viot processes, as established by Bertoin and Le Gall [9] and Dawson and Li [11]. 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 [33] and Lupu, Sabot and Tarr\`es [23] is shown to be valid in the general case.