The stochastic Jacobi flow

TL;DR

This paper introduces the Jacobi flow, a flow version of Jacobi processes, constructed via a common Brownian motion, and explores its connections with Fleming--Viot processes, Gaussian measures, and Bass--Burdzy flows.

Contribution

It provides a pathwise construction of the Jacobi flow for all parameters using a common Brownian motion, linking it to various stochastic processes and measure disintegrations.

Findings

Constructed Jacobi flow via perturbed reflecting Brownian motion.

Established connection between Jacobi flow and Fleming--Viot processes.

Unified framework for disintegrating Gaussian measures on the real line.

Abstract

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.