发表机构
Sorbonne Université; State Key Laboratory of Mathematical Sciences, AMSS, Chinese Academy of Sciences（索邦大学; 中国科学院数学与系统科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明Derrida-Retaux连续分支过程可由布朗运动编码，并通过时间反转和长度擦除从连续布朗树完全构造，从而揭示其关键量的统计规律与动力学本质。
AI 中文摘要
Derrida-Retaux连续分支过程被猜想为相应离散模型及许多其他临界层次重整化模型的标度极限，它是一个在时间区间$[0, 1)$上非均匀演化的细胞过程，其中每个细胞线性增长，并以特定速率独立分裂。在本文中，我们证明Derrida-Retaux连续分支过程一方面可由一族收敛到布朗运动的过程编码，另一方面可通过时间反转和长度擦除，完全由极限布朗路径编码的连续布朗树获得。这一直接表示解释了该模型中重要量（总质量、细胞数、大数定律）所遵循的特定规律，并提供了对其动力学的更好理解。
英文摘要
The Derrida-Retaux continuous branching process, which is conjectured to be the scaling limit of the corresponding discrete model and of many other critical hierarchical renormalization models, is a process of cells evolving inhomogeneously on the time interval $[0, 1)$ via linear growth of each cell and independent splitting of cells at a specific rate. In this article, we show that the Derrida-Retaux continuous branching process is, on the one hand, encoded by a family of processes converging to a Brownian motion and on the other hand, fully obtained from the continuum Brownian tree encoded by the limiting Brownian path by time reversal and through length erasure of this tree. This direct representation explains the specific laws governing the significant quantities featuring the model (total mass, number of cells, law of large numbers) and provides a better understanding of its dynamics.
Comments41 pages, 9 figures
