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- W4301731380 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 $X:$ $Pr(I_{t+s} = j | (X_u, I_u)_{u leq t}) = a_{i j}(X_t) s + o(s)$ for $i neq j$ 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 $Lambda^-$ and $Lambda^+$ respectively %called the {em minimal} and {em maximal average growth rate.} $Lambda^-$ (respectively $Lambda^+$) 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 $(Theta, J)$ 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, $X_t to p$ 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." @default.
- W4301731380 created "2022-10-05" @default.
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- W4301731380 date "2017-02-10" @default.
- W4301731380 modified "2023-10-16" @default.
- W4301731380 title "Random Switching between Vector Fields Having a Common Zero" @default.
- W4301731380 doi "https://doi.org/10.48550/arxiv.1702.03089" @default.
- W4301731380 hasPublicationYear "2017" @default.
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