Variable speed branching Brownian motion 1. Extremal processes in the weak correlation regime

TL;DR

This paper establishes the convergence of extremal processes in variable speed branching Brownian motions with specific speed functions, revealing universal limiting behavior dependent only on initial slope and final time.

Contribution

It extends previous two-speed BBM results to general variable speed cases using Gaussian comparison, under weak regularity conditions.

Findings

Proves convergence of extremal processes for variable speed BBM.

Shows universality of the limiting extremal process.

Extends prior results from two-speed to general variable speed scenarios.

Abstract

We prove the convergence of the extremal processes for variable speed branching Brownian motions where the "speed functions", that describe the time-inhomogeneous variance, lie strictly below their concave hull and satisfy a certain weak regularity condition. These limiting objects are universal in the sense that they only depend on the slope of the speed function at $0$ and the final time $t$. The proof is based on previous results for two-speed BBM obtained in a recent paper of ours and uses Gaussian comparison arguments to extend these to the general case.