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Asymptotic expansion of the invariant measure for Markov-modulated ODEs at high frequency

Part of: Markov processes Random dynamical systems Stochastic analysis

Published online by Cambridge University Press:  30 January 2025

Pierre Monmarché Open the ORCID record for Pierre Monmarché [Opens in a new window]  and
Edouard Strickler Open the ORCID record for Edouard Strickler [Opens in a new window]
Pierre Monmarché*
Affiliation:
Sorbonne Université
Edouard Strickler*
Affiliation:
Université de Lorraine, CNRS, Inria, IECL
*
*Postal address: 4 place Jussieu 75005 Paris, France. Email: pierre.monmarche@sorbonne-universite.fr
**Postal address: Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France. Email: edouard.strickler@univ-lorraine.fr
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Abstract

We consider time-inhomogeneous ordinary differential equations (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 a random environment. In particular, as a corollary, we prove the existence of two matrices whose convex combinations are all stable but are such that, for a suitable jump rate, the top Lyapunov exponent of a Markov-modulated linear ODE switching between these two matrices is positive.

Keywords

AveragingLyapunov exponentpiecewise deterministic Markov processcooperative linear system

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60H10: Stochastic ordinary differential equations
Secondary: 37H15: Multiplicative ergodic theory, Lyapunov exponents
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 27
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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