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

TL;DR

This paper derives an asymptotic expansion for the invariant measure of Markov-modulated ODEs at high frequency, providing explicit formulas and applying results to various models including Lotka-Volterra in random environments.

Contribution

It introduces a novel asymptotic expansion for the invariant measure of Markov-modulated ODEs as the Markov process accelerates, with explicit first-order terms and diverse applications.

Findings

Asymptotic expansion of invariant measure in high-frequency regime

Explicit first-order correction formula derived

Application to stability analysis in Markov-modulated systems

Abstract

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.