Transformation of Stochastic Recursions and Critical Phenomena in the Analysis of the Aldous-Shields-Athreya Cascade and Related Mean Flow Equations

TL;DR

This work develops a probabilistic framework on trees to study the Aldous–Shields–Athreya cascade and its connection to two mean-field-type equations, the pantograph equation $w'(t)=-w(t)+a w(\alpha t)$ and the $\alpha$-Riccati equation $u'(t)=-u(t)+u^2(\alpha t)$. By constructing self-similar cascades and employing stochastic transformations, the authors show that for $\alpha>1$ the $\alpha$-Riccati equation admits infinitely many global solutions for a range of initial data, with a precise asymptotic rate $\lim_{t\to\infty}(1-u_\lambda(t))/t^{-\gamma}=\lambda$, where $\gamma=\log_\alpha 2$. Central to the results are stochastic-Picard iterations, explosion/hyperexplosion phenomena, and a one-to-many principle that links a single unary pantograph solution to a whole family of multiplicative $\alpha$-Riccati solutions via $X_\lambda=e^{\lambda\mathscr{X}}$. The paper also provides algorithmic Monte Carlo schemes to illustrate the constructions and delineates how nonuniqueness emerges from the microstructure of the ASA cascade, with implications for nonlocal, nonlinear mean-field models such as those arising in fluid dynamics and turbulence.

Abstract

The paper has two main goals. First, we extend the contemporary probability theory on trees to investigate critical phenomena in a stochastic model of Yule type called Aldous-Shields-Athreya (ASA) cascade. Second, we apply the newly developed probabilistic framework to problems of uniqueness and nonuniqueness of solutions to the linear and nonlinear mean flow equations, referred to as the pantograph equation and $α$-Riccati equation, respectively. The stochastic processes associated with these equations are related to each other via a one-parameter family of transformations. Remarkably, these simple transformations lead to infinitely many solutions to the initial-value problem of the nonlinear mean flow equation. Despite being non-explicit at the level of mean flow, their effect on the mean flow equations is reminiscent of how the Cole-Hopf transformation maps solutions of the heat equation to those of the Burgers equation. While the ASA cascade has been used to model percolation, ageing, and data compression, its relevance to any specific physical molecular dynamics is unclear to the authors. Nevertheless, our results highlight how simple stochastic-level transformations can uncover significant macroscopic structures. This principle is exemplified by the connection between spontaneous magnetization and shocks in the Burgers equation (Newman 1986) or the connection between the branching Brownian motion and the KPP equation (McKean 1975). In our model, the breakdown of uniqueness in mean flow solutions corresponds to critical phenomena in the ASA cascade such as stochastic explosion, hyperexplosion, and percolation.

Figures (7)

• Figure 1: Pantograph unary tree
• Figure 2: Aldous-Shields-Athreya (ASA) cascade
• Figure 3: Graphs of $\eta_2$ (left) and of $\eta_3$ (right) with numerical artifacts ($\alpha=3,a=2$)
• Figure 4: Comparison of $\eta_0$,$\eta_1$,...,$\eta_6$ where $\alpha=3,a=2$
• Figure 5: $\alpha=3,a=2,K=15,\bar{K}=6$. Left figure: graph of $\tilde{\eta}_2$ and its Monte Carlo approximation. Right figure: graph of $\tilde{\eta}'_2\approx-\tilde{\eta}_2+a\tilde{\eta}_2(\alpha t)$ and its Monte Carlo approximation. Bottom figure: graph of $\int_0^t\tilde{\eta}_2(s)ds$ and its Monte Carlo approximation.

Theorems & Definitions (58)

• Remark 1
• Remark 2
• Remark 3
• Theorem 1: see athreyaalphariccatidascaliuc2023errata
• Remark 4
• Theorem 2
• proof
• Remark 5
• Corollary 1
• proof