The Annals of Applied Probability

Random switching between vector fields having a common zero

Michel Benaïm and Edouard Strickler

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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 $M\times E$ defined by $\dot{X}_{t}=F^{I_{t}}(X_{t})$ where $I$ is a jump process controlled by $X$: ${\mathsf{P}}(I_{t+s}=j|(X_{u},I_{u})_{u\leq t})=a_{ij}(X_{t})s+o(s)$ for $i\neq j$ on $\{I_{t}=i\}$.

We show that the behaviour of $(X,I)$ is mainly determined by the behaviour 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 $\Lambda^{-}$ and $\Lambda^{+}$, respectively, defined as the minimal (resp., maximal) growth rate of $\|Y_{t}\|$, where the minimum (resp., maximum) is taken over all the ergodic measures of the angular process $(\Theta,J)$ with $\Theta_{t}=\frac{Y_{t}}{\|Y_{t}\|}$. It is shown that $\Lambda^{+}$ coincides with the top Lyapunov exponent (in the sense of ergodic theory) of $(Y,J)$ and that under general assumptions $\Lambda^{-}=\Lambda^{+}$. We then prove that, under certain irreducibility conditions, $X_{t}\rightarrow p$ exponentially fast when $\Lambda^{+}<0$ and $(X,I)$ converges in distribution at an exponential rate toward a (unique) invariant measure supported by $M\setminus \{p\}\times E$ when $\Lambda^{-}>0$. Some applications to certain epidemic models in a fluctuating environment are discussed and illustrate our results.

Article information

Ann. Appl. Probab., Volume 29, Number 1 (2019), 326-375.

Received: September 2017
Revised: June 2018
First available in Project Euclid: 5 December 2018

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Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces 34A37: Differential equations with impulses 37H15: Multiplicative ergodic theory, Lyapunov exponents [See also 34D08, 37Axx, 37Cxx, 37Dxx] 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 92D30: Epidemiology

Piecewise deterministic Markov processes random switching Lyapunov exponents stochastic persistence Hörmander-bracket conditions epidemic models SIS


Benaïm, Michel; Strickler, Edouard. Random switching between vector fields having a common zero. Ann. Appl. Probab. 29 (2019), no. 1, 326--375. doi:10.1214/18-AAP1418.

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