Numerical bounds on the regularity of an invariant function: Probability of extinction of Galton-Watson processes in dynamical environments

TL;DR

The paper tackles the problem of quantifying the regularity of the extinction-probability invariant function $q$ for Galton–Watson processes in dynamical environments by linking Hölder smoothness and differentiability to the ratio $|\lambda_F|/\lambda_u$ of two Lyapunov exponents. It develops rigorous numerical methods to bound $\lambda_F$ and $\lambda_u$ from below and above, using periodic orbits, interval arithmetic, and careful handling of the unknown regularity of $q$, then applies these bounds to establish precise regularity results in uniformly supercritical regimes. The main contributions are (i) effective, certified algorithms for lower and upper bounds on the fibre and base Lyapunov exponents, (ii) a concrete procedure to bound the extinction probability $q$ and its derivatives, and (iii) a demonstration on explicit models showing how $q$’s Hölder regularity and differentiability class depend on the Lyapunov ratio. These results provide a rigorous bridge between dynamical-systems techniques and probabilistic extinction phenomena, enabling reliable regularity conclusions and quantitative estimates in complex, environment-dependent branching processes.

Abstract

We study the Lyapunov exponents of models that are close to skew product systems over a C__ uniformly expanding transformation of the circle. For a continuous fibre map $φ$, analytic, increasing, and convex in the fibre variable, we consider the smallest invariant function q satisfying q(x) = $φ$(x, q(T x)). We provide rigorous numerical bounds on two Lyapunov exponents (the fibre exponent and the base exponent), and present algorithms to compute these bounds effectively. We then apply this framework to Galton-Watson processes in dynamical environments in the uniformly supercritical case. The probability of extinction q of the process is the invariant function of the associated system. Using the previously computed Lyapunov exponents, we control the H{ö}lder regularity and differentiability class of the probability of extinction.

Figures (7)

• Figure 1: Plot of rigorous lower bounds on the function $x\in \mathbb{R}/\!\raisebox{-.65ex}{$\mathbb{Z}$}\mapsto\varphi^{(n)}_{\lambda,\omega}(x,0)$ in the case of Example \ref{['ex1']} with $N=2$ and $\varepsilon=0$ for different values of $n$.
• Figure 2: Plot of rigorous upper bounds on the function $x\in \mathbb{R}/\!\raisebox{-.65ex}{$\mathbb{Z}$}\mapsto\varphi^{(n)}_{\lambda,\omega}(x,K)$ in the case of Example \ref{['ex1']} with $N=2$ and $\varepsilon=0$ for different values of $n\in\mathbb{N}$. The lower bound on $q$ is obtained by displaying the function $x\in \mathbb{R}/\!\raisebox{-.65ex}{$\mathbb{Z}$}\mapsto\varphi^{(10)}_{\lambda,\omega}(x,0)$.
• Figure 3: Plot of the logarithm of the absolute value of the Fourier coefficients of an approximation of $q_\lambda$ for coefficients between $-100$ and $100$ in the case of Example \ref{['ex1']} with $N=2$, $\varepsilon=0$, $\lambda=0.55$, and $\omega=0$.
• Figure 4: Plot of rigorous upper bounds on $\frac{|\lambda_F|}{\lambda_u}$ obtained by considering only orbits of primitive period $M$ as a function of $\omega\in \mathbb{R}/\!\raisebox{-.65ex}{$\mathbb{Z}$}$ in the case of Example \ref{['ex1']} with $N=2$, $\varepsilon=0$, and $\lambda=1.05$. The lower bound on $\frac{|\lambda_F|}{\lambda_u}$ is obtained thanks to Algorithm \ref{['alg_ub']}.
• Figure 5: Plot of rigorous bounds on $\frac{|\lambda_F|}{\lambda_u}$ as a function of $\lambda\in\mathbb{R}$ in the case of Example \ref{['ex1']} with $N=2$, $\varepsilon=0$, and $\omega=0$.

Theorems & Definitions (35)

• Remark 1.0.1
• Theorem 1.0.2
• Definition 2.2.1
• Proposition 2.2.2
• Definition 2.2.3
• Proposition 2.2.4
• Definition 2.2.5
• Theorem 2.3.1
• Theorem 2.3.2
• Remark 3.0.1