Ergodic behavior of products of random positive operators

TL;DR

The paper develops an ergodic theory for products of random positive operators acting on signed measures in infinite-dimensional spaces, extending classical finite-dimensional results to non-conservative, non-compact settings. It introduces an inhomogeneous Doeblin contraction via auxiliary Markov operators to obtain a geometric decomposition $\mu M_{0,n} \simeq \mu(h) r_n \pi_n$ with a random Lyapunov exponent $\lambda$ governing the growth of $r_n$, and a weakly converging sequence of random probability measures $\pi_n$ to a limit $\Lambda$. In the i.i.d. case, it links $\lambda$ to an average over $\Lambda$ and characterizes oscillations when $\lambda=0$ (with a Null-Homology caveat). The framework is applied to infinite Leslie matrices, yielding practical conditions for population dynamics models with infinitely many types and illustrating both the reach and limitations of the approach. These results provide a foundation for analyzing MGWRE and other infinite-type population processes under random environmental fluctuations.

Abstract

This article is devoted to the study of products of random operators of the form $M_{0,n}=M_0\cdots M_{n-1}$, where $(M_{n})_{n\in\mathbb{N}}$ is an ergodic sequence of positive operators on the space of signed measures on a space $\mathbb{X}$. Under suitable conditions, in particular, a Doeblin-type minoration suited for non conservative operators, we obtain asymptotic results of the form \[ μM_{0,n} \simeq μ(\tilde{h}) r_n π_n,\] where $\tilde{h}$ is a random bounded function, $(r_n)_{n\geq 0}$ is a random non negative sequence and $π_n$ is a random probability measure on $\mathbb{X}$. Moreover, $\tilde{h}$, $(r_n)$ and $π_n$ do not depend on the choice of the measure $μ$. We prove additionally that $n^{-1} \log (r_n)$ converges almost surely to the Lyapunov exponent $λ$ of the process $(M_{0,n})_{n\geq 0}$ and that the sequence of random probability measures $(π_n)$ converges weakly towards a random probability measure. These results are analogous to previous estimates from Hennion in the case of $d\times d$ matrices, that were obtained with different techniques, based on a projective contraction in Hilbert distance. In the case where the sequence $(M_n)$ is i.i.d, we additionally exhibit an expression of the Lyapunov exponent $λ$ as an integral with respect to the weak limit of the sequence of random probability measures $(π_n)$ and exhibit an oscillation behavior of $r_n$ when $λ=0$. We provide a detailed comparison of our assumptions with the ones of Hennion and present some example of applications of our results, in particular in the field of population dynamics.

Theorems & Definitions (46)

• Remark 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 3.1
• Lemma 3.2
• proof : Proof of Lemma \ref{['lem:Doeblin']}
• proof : Proof of Proposition \ref{['prop: contraction']}
• Lemma 3.3
• proof : Proof of Lemma \ref{['prop:error terms']}