Random Interlacements via Kuznetsov Measures

TL;DR

This paper presents an alternative probabilistic potential theory-based construction of random interlacements, connecting them to Kuznetsov measures and two-sided Brownian motions or random walks, providing new insights into their structure.

Contribution

It introduces a novel construction of random interlacements using Kuznetsov measures and classical potential theory, linking them to two-sided stochastic processes.

Findings

Interlacements can be constructed via Kuznetsov measures and potential theory.

Random interlacements form a Poisson cloud of two-sided Brownian motions or random walks.

The construction involves processes started in Lebesgue measure and conditioned on proximity to the origin.

Abstract

The aim of this note is to give an alternative construction of interlacements - as introduced by Sznitman - which makes use of classical probabilistic potential theory. In particular, we outline that the intensity measure of an interlacement is known in probabilistic potential theory under the name "approximate Markov chain" or "quasi-process". We provide a simple construction of random interlacements through (unconditioned) two-sided Brownian motions (resp. two-sided random walks) involving Mitro's general construction of Kuznetsov measures and a Palm measures relation due to Fitzsimmons. In particular, we show that random interlacement is a Poisson cloud (`soup') of two-sided random walks (or Brownian motions) started in Lebesgue measure and restricted on being closest to the origin at time between 0 and 1 - modulus time-shift.