The fixed points of branching Brownian motion

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 $$\theta $$ is locally finite and $$\theta ({\mathbb {R}}_+)<\infty $$ almost surely.