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February 2019 Random switching between vector fields having a common zero
Michel Benaïm, Edouard Strickler
Ann. Appl. Probab. 29(1): 326-375 (February 2019). DOI: 10.1214/18-AAP1418

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 $M\times E$ 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.

Citation

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Michel Benaïm. Edouard Strickler. "Random switching between vector fields having a common zero." Ann. Appl. Probab. 29 (1) 326 - 375, February 2019. https://doi.org/10.1214/18-AAP1418

Information

Received: 1 September 2017; Revised: 1 June 2018; Published: February 2019
First available in Project Euclid: 5 December 2018

zbMATH: 07039127
MathSciNet: MR3910006
Digital Object Identifier: 10.1214/18-AAP1418

Subjects:
Primary: 34A37, 37A50, 37H15, 60J25, 92D30

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 1 • February 2019
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