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 under the only assumption that $\theta ({\mathbb{R}}_{+})<\mathrm{\infty }$ almost surely.