A one-parameter family of local-time conditioned processes : the generalized Brownian burglars

Abstract

We construct and characterize the one-parameter family of real-valued processes that allow to recover the law of a Brownian loop-soup of any intensity on the real line conditionally on its occupation time field. These processes generalize the Brownian burglar constructed by Warren and Yor (that corresponds to our process in the limiting case where the intensity vanishes and there is just one Brownian motion). These processes are closely related to the Bass-Burdzy flow and recent work of A\"id\'ekon, Hu and Shi on the stochastic Jacobi flow. Our approach uses a formalism, building on the notion of driver processes, where one considers the evolving object to be the line that we ``stretch", instead of the burglar itself and its remaining occupation time. This leads to a characterization of these (generalized) Brownian burglars by simple natural axioms. It also allows to treat ``negative intensities'' and provides fairly direct derivations of several properties of the burglars, and in particular a ``target-independence/locality" property at the special intensity where the local time is the square of a Gaussian Free Field.