Entrance laws for annihilating Brownian motions and the continuous-space voter model

TL;DR

This paper classifies all possible initial configurations (entrance laws) for systems of annihilating Brownian motions starting from dense sets, by linking them to the interface model of the continuous-space voter model and analyzing their properties.

Contribution

It provides a complete classification of entrance laws for annihilating Brownian motions using their connection to the continuous-space voter model.

Findings

Full classification of entrance laws for aBMs.

Examples of finite approximations leading to different entrance laws.

Expression for the n-point densities of aBMs from arbitrary entrance laws.

Abstract

Consider a system of particles moving independently as Brownian motions until two of them meet, when the colliding pair annihilates instantly. The construction of such a system of annihilating Brownian motions (aBMs) is straightforward as long as we start with a finite number of particles, but is more involved for infinitely many particles. In particular, if we let the set of starting points become increasingly dense in the real line it is not obvious whether the resulting systems of aBMs converge and what the possible limit points (entrance laws) are. In this paper, we show that aBMs arise as the interface model of the continuous-space voter model. This link allows us to provide a full classification of entrance laws for aBMs. We also give some examples showing how different entrance laws can be obtained via finite approximations. Further, we discuss the relation of the continuous-space voter model to the stepping stone and other related models. Finally, we obtain an expression for the $n$-point densities of aBMs starting from an arbitrary entrance law.