2023/09/28 by Monmarché, Pierre, Strickler, Edouard · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2309.16464
We consider time-inhomogeneous ODEs whose parameters are governed by an underlying ergodic Markov process. When this underlying process is accelerated by a factor ε-1, an averaging phenomenon occurs and the solution of the ODE converges to a deterministic ODE as ε 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 ε 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.