[Paper Review] Random Switching between Vector Fields Having a Common Zero

Michel Benaı̈m, Édouard Strickler|arXiv (Cornell University)|Feb 10, 2017

Mathematical Dynamics and Fractals6 references4 citations

TL;DR

This paper studies piecewise deterministic Markov processes (PDMPs) that randomly switch between vector fields sharing a common zero at the origin. By analyzing the linearized system around the equilibrium, it establishes that the top Lyapunov exponent determines long-term behavior: if negative, the process converges exponentially to the origin; if positive, it converges in distribution to a unique invariant measure supported away from the origin, under irreducibility and hypoellipticity conditions.

ABSTRACT

Let $E$ be a finite set, $\{F^i\}_{i \in E}$ a family of vector fields on $\mathbb{R}^d$ leaving positively invariant a compact set $M$ and having a common zero $p \in M.$ We consider a piecewise deterministic Markov process $(X,I)$ on $M imes E$ defined by $\dot{X}_t = F^{I_t}(X_t)$ where $I$ is a jump process controlled by $X:$ $\Pr(I_{t+s} = j | (X_u, I_u)_{u \leq t}) = a_{i j}(X_t) s + o(s)$ for $i eq j$ on $\{I_t = i \}.$ We show that the behavior of $(X,I)$ is mainly determined by the behavior of the linearized process $(Y,J)$ where $\dot{Y}_t = A^{J_t} Y_t,$ $A^i$ is the Jacobian matrix of $F^i$ at $p$ and $J$ is the jump process with rates $(a_{ij}(p)).$ We introduce two quantities $Λ^-$ and $Λ^+$ respectively %called the {\em minimal} and {\em maximal average growth rate.} $Λ^-$ (respectively $Λ^+$) is defined as the {\em minimal} (respectively {\em maximal}) {\em growth rate} of $\|Y_t\|,$ where the minimum (respectively maximum) is taken over all the ergodic measures of the angular process $(Θ, J)$ with $Θ_t = \frac{Y_t}{\|Y_t\|}.$ It is shown that $Λ^+$ coincides with the top Lyapunov exponent (in the sense of ergodic theory) of $(Y,J)$ and that under general assumptions $Λ^- = Λ^+.$ We then prove that, under certain irreducibility conditions, $X_t o p$ exponentially fast when $Λ^+ < 0$ and $(X,I)$ converges in distribution at an exponential rate toward a (unique) invariant measure supported by $M \setminus \{p\} imes E$ when $Λ^- > 0.$ Some applications to certain epidemic models in a fluctuating environment are discussed and illustrate our results.

Motivation & Objective

• To understand the long-term behavior of a PDMP that switches randomly between vector fields sharing a common equilibrium at the origin.
• To determine under what conditions the process converges to the equilibrium (extinction) or persists away from it (stochastic persistence).
• To establish a connection between the top Lyapunov exponent of the linearized system and the asymptotic behavior of the nonlinear system.
• To extend results on exponential convergence and invariant measures to systems with state-dependent switching rates and compact invariant sets.
• To apply the framework to epidemic models in fluctuating environments, illustrating persistence and extinction scenarios.

Proposed method

• Formalize the PDMP dynamics as $\dot{X}_t = F^{I_t}(X_t)$, where $I_t$ is a jump process with state-dependent rates $a_{ij}(x)$.
• Linearize the system around the common zero $p=0$, yielding $\dot{Y}_t = A^{J_t} Y_t$, with $A^i$ the Jacobian of $F^i$ at 0.
• Define the angular process $\Theta_t = Y_t / \|Y_t\|$, and analyze its ergodic measures to define growth rates $\Lambda^-$ and $\Lambda^+$.
• Show that $\Lambda^+$ equals the top Lyapunov exponent of the linearized system $(Y,J)$, and under general conditions, $\Lambda^- = \Lambda^+$.
• Use stochastic persistence theory to prove convergence to an invariant measure when $\Lambda^- > 0$, and almost sure convergence to the origin when $\Lambda^+ < 0$.
• Apply the results to epidemic models with fluctuating environments, demonstrating how switching rates affect disease persistence or extinction.

Experimental results

Research questions

• RQ1 Under what conditions does the PDMP converge almost surely and exponentially fast to the common zero of the vector fields?
• RQ2 How does the top Lyapunov exponent of the linearized system determine the long-term behavior of the nonlinear switching process?
• RQ3 When does the process exhibit stochastic persistence, i.e., converge in distribution to a unique invariant measure supported on $M \setminus \{0\} \times E$?
• RQ4 What role do irreducibility and hypoellipticity (Hörmander-type) conditions play in ensuring exponential convergence and uniqueness of the invariant measure?
• RQ5 How can the framework be applied to model disease dynamics in randomly switching environments, and what determines persistence or extinction?

Key findings

• The top Lyapunov exponent $\Lambda^+$ of the linearized system $\dot{Y}_t = A^{J_t} Y_t$ coincides with the maximal growth rate $\Lambda^+$ of $\|Y_t\|$ over ergodic measures of the angular process.
• Under general assumptions, the minimal and maximal growth rates satisfy $\Lambda^- = \Lambda^+$, implying a unique exponential growth rate for the linearized system.
• If $\Lambda^+ < 0$, then $X_t \to 0$ almost surely and exponentially fast, indicating extinction of the process near the origin.
• If $\Lambda^- > 0$, then the process $(X_t, I_t)$ converges in distribution to a unique invariant probability measure supported on $M \setminus \{0\} \times E$, indicating stochastic persistence.
• Exponential convergence in total variation toward the invariant measure is established under irreducibility and hypoellipticity conditions.
• Applications to epidemic models show that switching between environmental states can lead to persistence or extinction depending on the sign of $\Lambda^+$, even when individual vector fields are unstable.