Entrance laws for coalescing and annihilating Brownian motions

TL;DR

This paper studies the initial conditions (entrance laws) of coalescing and annihilating Brownian motions on the line, showing that their extreme entrance laws are Pfaffian point processes with identifiable kernels.

Contribution

It characterizes the extreme entrance laws for these processes as Pfaffian point processes and explicitly identifies their kernels, advancing understanding of their initial distributions.

Findings

Extreme entrance laws are Pfaffian point processes.

Kernels of these Pfaffian processes are explicitly identified.

Provides a comprehensive description of initial conditions for coalescing and annihilating Brownian motions.

Abstract

Systems of instantaneously annihilating or coalescing Brownian motions on the line are considered. The extreme points of the set of entrance laws for this process are shown to be Pfaffian point processes at all times and their kernels are identified.