The fixed points of branching Brownian motion

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