Freezing and decorated Poisson point processes

Understand

The limiting extremal processes of the branching Brownian motion (BBM), the two-speed BBM, and the branching random walk are known to be randomly shifted decorated Poisson point processes (SDPPP).

• In the proofs of those results, the Laplace functional of the limiting extremal process is shown to satisfy $L[\theta_y f]=g(y-\tau_f)$ for any nonzero, nonnegative, compactly supported, continuous function $f$, where $\theta_y$ is the shift operator, $\tau_f$ is a real number that depends on $f$, and $g$ is a real function that is independent of $f$.
• We show that, under some assumptions, this property characterizes the structure of SDPPP.
• Moreover, when it holds, we show that $g$ has to be a convolution of the Gumbel distribution with some measure.