Fluctuations of additive martingale limits of branching Brownian motion
Abstract
Consider a one-dimensional branching Brownian motion. Let W∞(β) denote the limit of the additive martingale in the subcritical regime lvert βrvert < βc and Z∞ be the limit of the derivative martingale at criticality. Madaule (\textit{Stochastic Process. Appl.} 126 (2016), no. 2, 470--502) established the following convergence W∞(β)/(βc−β)xrightarrow[βnearrow βc]ℙ 2Z∞. The goal of this paper is twofold: firstly, we strengthen this result into an almost sure convergence; secondly, we describe the fluctuations occurring in this convergence by proving 1/(βc−β)( W∞(β)/(βc−β) − 2 Z∞ +2(βc−β)log(βc−β) Z∞) xrightarrow[βnearrow βc](d) S, where, conditionally on Z∞, S follows a spectrally negative 1-stable distribution with scale and shift parameters proportional to Z∞. Furthermore, these results are extended to the setting of complex additive martingales and the fluctuations to a multi-dimensional convergence.
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BibTeX
@misc{chen2026fluctuations,
title = {{Fluctuations of additive martingale limits of branching Brownian motion}},
author = {Xinxin Chen and Michel Pain},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10530},
eprint = {2609.10530},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05745098v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fluctuations of additive martingale limits of branching Brownian motion
AU - Xinxin Chen
AU - Michel Pain
PY - 2026
DA - 2026/09/09
PB - HAL
DO - 10.48550/arxiv.2609.10530
UR - https://hal.science/hal-05745098v1
AB - Consider a one-dimensional branching Brownian motion. Let W_∞(β) denote the limit of the additive martingale in the subcritical regime lvert βrvert < β_c and Z_∞ be the limit of the derivative martingale at criticality. Madaule (\textit{Stochastic Process. Appl.} 126 (2016), no. 2, 470--502) established the following convergence W_∞(β)/(β_c−β)xrightarrow[βnearrow β_c]ℙ 2Z_∞. The goal of this paper is twofold: firstly, we strengthen this result into an almost sure convergence; secondly, we describe the fluctuations occurring in this convergence by proving 1/(β_c−β)( W_∞(β)/(β_c−β) − 2 Z_∞ +2(β_c−β)log(β_c−β) Z_∞) xrightarrow[βnearrow β_c](d) S, where, conditionally on Z_∞, S follows a spectrally negative 1-stable distribution with scale and shift parameters proportional to Z_∞. Furthermore, these results are extended to the setting of complex additive martingales and the fluctuations to a multi-dimensional convergence.
ER -