Two-dimensional Brownian random interlacements

TL;DR

This paper introduces a new model of two-dimensional continuous random interlacements based on Brownian trajectories, providing insights into the local behavior of Wiener sausages and connecting to existing discrete models.

Contribution

It develops the first continuous analogue of 2D discrete random interlacements using Brownian motion conditioned on avoiding sets, extending the theory to continuous settings.

Findings

Established properties analogous to discrete interlacements

Connected Brownian loops to the interlacement model

Provided results specific to the continuous case

Abstract

We introduce the model of two-dimensional continuous random interlacements, which is constructed using the Brownian trajectories conditioned on not hitting a fixed set (usually, a disk). This model yields the local picture of Wiener sausage on the torus around a late point. As such, it can be seen as a continuous analogue of discrete two-dimensional random interlacements [Comets, Popov, Vachkovskaia, 2016]. At the same time, one can view it as (restricted) Brownian loops through infinity. We establish a number of results analogous to these of [Comets, Popov, Vachkovskaia, 2016; Comets, Popov, 2016], as well as the results specific to the continuous case.