[Paper Review] Poisson ensembles of loops of one-dimensional diffusions

Titus Lupu|HAL (Le Centre pour la Communication Scientifique Directe)|Feb 15, 2013

Mathematical Dynamics and Fractals10 citations

TL;DR

This paper establishes a comprehensive framework for Poisson ensembles of unrooted loops in one-dimensional diffusions, leveraging time and scale changes, h-transforms, and loop measures. It proves that the occupation field of such ensembles at intensity 1/2 equals half the square of a Gaussian free field, extending Dynkin's isomorphism, and shows that loop-erased random walks generate Poisson ensembles of loops at intensity 1, generalizing Wilson's algorithm.

ABSTRACT

We study the analogue of Poisson ensembles of Markov loops ('loop soups') in the setting of one-dimensional diffusions. We give a detailed description of the corresponding intensity measure. The properties of this measure on loops lead us to an extension of Vervaat's bridge-to-excursion transformation that relates the bridges conditioned by their minimum and the excursions of all the diffusion we consider and not just the Brownian motion. Further we describe the Poisson point process of loops, their occupation fields and explain how to sample these Poisson ensembles of loops using two-dimensional Markov processes. Finally we introduce a couple of interwoven determinantal point processes on the line which is a dual through Wilson's algorithm of Poisson ensembles of loops and study the properties of these determinantal point processes.

Motivation & Objective

• To develop a self-contained theory of loop measures and Poisson ensembles for one-dimensional diffusions.
• To extend the universal properties of loop ensembles—especially the isomorphism with the Gaussian free field and loop-erasure connection—to general diffusions.
• To establish invariance of loop measures under time-change, scale change, and conjugation by positive functions (h-transforms).
• To generalize Vervaat's bridge-to-excursion transformation to diffusions via conditioning on loop minima.
• To analyze loop measures with killing or creation of mass terms under negativity constraints.

Proposed method

• Uses unrooted loop measures defined via integral over time and space of transition densities and bridge measures.
• Applies time-change and scale transformation to show covariance of loop measures under speed and scale changes of the diffusion.
• Introduces a disintegration of the loop measure by conditioning on the minimum value of the loop, leading to a decomposition into excursion measures.
• Applies the h-transform (conjugation of generators) to relate different diffusions and prove uniqueness of loop measures up to conjugation.
• Uses Green's functions and speed measures to characterize equivalence of loop measures across different generators.
• Applies the loop-erasure algorithm to show that erased loops form a Poisson ensemble at intensity 1, generalizing results from Brownian motion.

Experimental results

Research questions

• RQ1 How can loop measures for one-dimensional diffusions be defined and characterized in a way that preserves invariance under time and scale changes?
• RQ2 What is the relationship between the occupation field of a Poisson ensemble of loops and the Gaussian free field in the one-dimensional setting?
• RQ3 How does the loop-erasure process on a diffusion path relate to a Poisson ensemble of loops, and what intensity parameter governs this?
• RQ4 Can the Vervaat bridge-to-excursion transformation be generalized to diffusions beyond Brownian motion?
• RQ5 Under what conditions on the generator does the loop measure remain invariant under conjugation by positive functions?

Key findings

• The occupation field of a Poisson ensemble of loops at intensity 1/2 has the same law as half the square of a Gaussian free field, extending Dynkin's isomorphism to diffusions.
• Loop-erased random walks on a diffusion generate a Poisson ensemble of loops at intensity 1, generalizing the Brownian loop-soup result.
• Two diffusions have identical loop measures if and only if their generators are related by conjugation via a positive continuous function u with d²u/dx² a signed measure and Lu ≤ 0.
• The loop measure is invariant under time-change by the inverse of a continuous additive functional, which underlies conformal invariance in two dimensions.
• The disintegration of the loop measure by minimum value yields a decomposition into excursion measures, generalizing Vervaat's transformation to diffusions.
• The measure on loops is preserved under killing or creation of mass terms as long as the creation term satisfies a negativity condition.