Asymptotic expansion of the invariant measurefor Markov-modulated ODEs at high frequency
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| Format: | Preprint |
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2023
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| _version_ | 1866913985956151296 |
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| author | Monmarché, Pierre Strickler, Edouard |
| author_facet | Monmarché, Pierre Strickler, Edouard |
| contents | We consider time-inhomogeneous ODEs whose parameters are governed by an underlying ergodic Markov process. When this underlying process is accelerated by a factor $\varepsilon^{-1}$, an averaging phenomenon occurs and the solution of the ODE converges to a deterministic ODE as $\varepsilon$ vanishes. We are interested in cases where this averaged flow is globally attracted to a point. In that case, the equilibrium distribution of the solution of the ODE converges to a Dirac mass at this point. We prove an asymptotic expansion in terms of $\varepsilon$ for this convergence, with a somewhat explicit formula for the first order term. The results are applied in three contexts: linear Markov-modulated ODEs, randomized splitting schemes, and Lotka-Volterra models in random environment. In particular, as a corollary, we prove the existence of two matrices whose convex combinations are all stable but such that, for a suitable jump rate, the top Lyapunov exponent of a Markov-modulated linear ODE switching between these two matrices is positive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_16464 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotic expansion of the invariant measurefor Markov-modulated ODEs at high frequency Monmarché, Pierre Strickler, Edouard Probability Dynamical Systems We consider time-inhomogeneous ODEs whose parameters are governed by an underlying ergodic Markov process. When this underlying process is accelerated by a factor $\varepsilon^{-1}$, an averaging phenomenon occurs and the solution of the ODE converges to a deterministic ODE as $\varepsilon$ vanishes. We are interested in cases where this averaged flow is globally attracted to a point. In that case, the equilibrium distribution of the solution of the ODE converges to a Dirac mass at this point. We prove an asymptotic expansion in terms of $\varepsilon$ for this convergence, with a somewhat explicit formula for the first order term. The results are applied in three contexts: linear Markov-modulated ODEs, randomized splitting schemes, and Lotka-Volterra models in random environment. In particular, as a corollary, we prove the existence of two matrices whose convex combinations are all stable but such that, for a suitable jump rate, the top Lyapunov exponent of a Markov-modulated linear ODE switching between these two matrices is positive. |
| title | Asymptotic expansion of the invariant measurefor Markov-modulated ODEs at high frequency |
| topic | Probability Dynamical Systems |
| url | https://arxiv.org/abs/2309.16464 |