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Averaging principle for jump processes depending on fast ergodic dynamics

Vincent Kagan, Edouard Strickler, Denis Villemonais

TL;DR

We address the averaging principle for a slow index process driven by a fast ergodic dynamic that accelerates over time, producing a two-time-scale system with jumps. By decomposing the state space into ergodic components and employing a piecing-out construction, we show that the slow index process converges to an autonomous pure-jump process on the index set, with explicit jump timing and transition structure given by $G(t\mu_i(b))$ and $P(i,J)=\int_E \frac{\mu_i(dx)}{\mu_i(b)} b(x) \nu(x,J)$. The results yield concrete limiting dynamics for applications such as typed branching processes converging to continuous-time Galton–Watson processes and epidemic models with fast viral-load dynamics converging to classical contact processes. The framework extends classical averaging to non-Markov, ergodic fast components and provides a rigorous basis for replacing complex multiscale models with tractable pure-jump limits, while clarifying explosion properties and convergence in Skorokhod topology. Overall, the work delivers a principled reduction method for two-time-scale jump systems with ergodic fast dynamics and demonstrates its utility in biological and network context modeling.

Abstract

We consider a slow-fast stochastic process where the slow component is a jump process on a measurable index set whose transition rates depend on the position of the fast component. Between the jumps, the fast component evolves according to an ergodic dynamic in a state space determined by the index process. We prove that, when the ergodic dynamics are accelerated, the slow index process converges to an autonomous pure jump process on the index set. We apply our results to prove the convergence of a typed branching process toward a continuous-time Galton-Watson process, and of an epidemic model with fast viral loads dynamics to a standard contact process.

Averaging principle for jump processes depending on fast ergodic dynamics

TL;DR

We address the averaging principle for a slow index process driven by a fast ergodic dynamic that accelerates over time, producing a two-time-scale system with jumps. By decomposing the state space into ergodic components and employing a piecing-out construction, we show that the slow index process converges to an autonomous pure-jump process on the index set, with explicit jump timing and transition structure given by and . The results yield concrete limiting dynamics for applications such as typed branching processes converging to continuous-time Galton–Watson processes and epidemic models with fast viral-load dynamics converging to classical contact processes. The framework extends classical averaging to non-Markov, ergodic fast components and provides a rigorous basis for replacing complex multiscale models with tractable pure-jump limits, while clarifying explosion properties and convergence in Skorokhod topology. Overall, the work delivers a principled reduction method for two-time-scale jump systems with ergodic fast dynamics and demonstrates its utility in biological and network context modeling.

Abstract

We consider a slow-fast stochastic process where the slow component is a jump process on a measurable index set whose transition rates depend on the position of the fast component. Between the jumps, the fast component evolves according to an ergodic dynamic in a state space determined by the index process. We prove that, when the ergodic dynamics are accelerated, the slow index process converges to an autonomous pure jump process on the index set. We apply our results to prove the convergence of a typed branching process toward a continuous-time Galton-Watson process, and of an epidemic model with fast viral loads dynamics to a standard contact process.
Paper Structure (20 sections, 22 theorems, 208 equations, 4 figures)

This paper contains 20 sections, 22 theorems, 208 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Base process and first properties
  3. Dynamic of $\mathbf X^1$ between the jumps of $\mathbf I^1$
  4. Law of the jump time of $\mathbf{I}^1$
  5. Transition kernel
  6. Main results
  7. Accelerated and index processes
  8. Convergence of the index process
  9. Explosion/non-explosion properties
  10. Convergence in law of the index process in Skorokhod topology
  11. Comments on Assumption \ref{['assumption:ergo']} and \ref{['hyp:b-UI']}: Positive Harris recurrence
  12. Applications
  13. Branching processes
  14. Generalized contact process
  15. Proofs of the convergence results at jump times
  16. ...and 5 more sections

Key Result

Proposition 2.1

There exists a random process $\mathbf{X}^1 = (\Omega,\mathcal{F}^1,(\mathbf{X}_t^1)_{t\geq0}, (\mathbf{P}_x^1)_{x\in E})$ with state space $E\cup \{\Delta\}$ with lifetime $T_\Delta^1:=\inf\{t\geq 0,\mathbf X_t^1=\Delta\}$, and a random sequence of times $(\tau_k^1)_{k \geq 0}$ such that In addition, this process satisfies the strong Markov property at its transition times $(\tau_k^1)_{k \geq 0}

Figures (4)

  • Figure 1: Schematic representation of the dynamic of $\mathbf X^1$, built by piecing out and evolving according to specified stochastic dynamics on each $E_i$, $i\in I$.
  • Figure 2: Graphical representation of the state space of Example \ref{['exa:31']}
  • Figure 3: Infection rates and distribution of viral load of a newly infected individual, with the notation $q(x_\ell,\cdot)=\frac{\kappa_{0\to 1}(x_\ell)}{\sum_{j\sim k}i_j\,\kappa_{0 \to 1}(x_{j})}\,p(x_\ell,\,\cdot\,)$. The individual in the center is numbered $(0,0)$.
  • Figure 4: Healing of a previously infected individual. The individual in the center is numbered $(0,0)$.

Theorems & Definitions (58)

  • Remark 2.1
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 3.1
  • Proposition 3.1
  • ...and 48 more