From 1 to infinity: The log-correction for the maximum of variable-speed branching Brownian motion

TL;DR

This work analyzes the extreme-value behavior of variable-speed branching Brownian motion (VSBBM) when the time-dependent variance converges to the identity. By separating speed functions into Case A (above identity) and Case B (below), it derives explicit centering terms $m^+(t)$ and $m^-(t)$ and proves that the maximum converges to a universal Law of the form ${\mathbb E}[\exp(-C Z e^{-\,\sqrt{2} y})]$, with $Z$ the derivative martingale limit. It further establishes that the extremal process converges to a decorated Poisson cascade, with atoms from a Poisson process and i.i.d. decorations; the log-corrections exhibit tunable prefactors depending on the speed profile, interpolating between known BBM and independent-particle limits. The analysis combines barrier localisation for extremal paths, FKPP-tail asymptotics, derivative-martingale techniques, and Gaussian comparison to extend results from piecewise-linear speeds to general speed profiles, thereby generalising Bovier–Hartung results to the multi-speed setting. The findings deepen understanding of how time-inhomogeneous variances shape extremes in log-correlated systems with potential applications to CREM/GREM-type models and related stochastic PDEs.

Abstract

We study the extremes of variable speed branching Brownian motion (BBM) where the time-dependent "speed functions", which describe the time-inhomogeneous variance, converge to the identity function. We consider general speed functions lying strictly below their concave hull and piecewise linear, concave speed functions. In the first case, the log-correction for the order of the maximum depends only on the rate of convergence of the speed function near 0 and 1 and exhibits a smooth interpolation between the correction in the i.i.d. case, $\frac{1}{2\sqrt{2}} \ln t$, and that of standard BBM, $\frac{3}{2\sqrt{2}} \ln t$. In the second case, we describe the order of the maximum in dependence of the form of speed function and show that any log-correction larger than $\frac{3}{2\sqrt{2}} \ln t$ can be obtained. In both cases, we prove that the limiting law of the maximum and the extremal process essentially coincide with those of standard BBM, using a first and second moment method which relies on the localisation of extremal particles. This extends the results of Bovier and Hartung for two-speed BBM.

Figures (2)

• Figure 1: Localisation of an extremal particle of three-speed BBM in Case A. The dashed line depicts the effect of entropic repulsion.
• Figure 2: Localisation of an extremal particle of three-speed BBM in Case B. The dotted line depicts the function $\sqrt{2} A_t(s/t) t$ for $s \in [0, t]$. The dashed box depicts fluctuations of order $\sqrt{ \min\{ A_t (s/t) t, t - A_t(s/t) t\} }$. The dashed line depicts the effect of entropic repulsion.

Theorems & Definitions (54)

• Theorem 1.3
• Theorem 1.4
• Lemma 1.5: Berestycki, Lemma 4.4
• Proposition 3.1: B_C, 1to6
• Lemma 3.2: B_C, Proposition 8.2
• Lemma 3.3
• proof
• Lemma 3.4: Slepian's Lemma
• proof
• Lemma 3.5: BH14.1, Lemma 2.2