The fixed points of Branching Brownian Motion

TL;DR

This paper characterizes all point processes on the real line that remain invariant under critical branching Brownian motion with drift, assuming finiteness on the positive real axis, advancing understanding of invariant measures in stochastic processes.

Contribution

It provides a complete characterization of invariant point processes for critical branching Brownian motion with drift, under minimal assumptions.

Findings

Identifies all invariant point processes under the specified conditions.

Establishes invariance characterization with minimal assumptions.

Advances understanding of invariant measures in branching processes.

Abstract

In this work, we characterize all the point processes $\theta=\sum_{i\in \mathbb{N}} \delta_{x_i}$ on $\mathbb{R}$ which are left invariant under branching Brownian motions with critical drift $-\sqrt{2}$. Our characterization holds under the only assumption that $\theta(\mathbb{R}_+)<\infty$ almost surely.