Reducing metastable continuous-space Markov chains to Markov chains on a finite set

Abstract

We consider continuous-space, discrete-time Markov chains on ${\mathbb{R}}^d$, that admit a finite number $N$ of metastable states. Our main motivation for investigating these processes is to analyse random Poincaré maps, which describe random perturbations of ordinary differential equations admitting several periodic orbits. We show that under a few general assumptions, which hold in many examples of interest, the kernels of these Markov chains admit $N$ eigenvalues exponentially close to $1$, which are separated from the remainder of the spectrum by a spectral gap that can be quantified. Our main result states that these Markov chains can be approximated, uniformly in time, by a finite Markov chain with $N$ states. The transition probabilities of the finite chain are exponentially close to first-passage probabilities at neighbourhoods of metastable states, when starting in suitable quasistationary distributions.

arXiv commentAccepted version for Annales de l'Institut Henri Poincaré. 46 pages, minor changes (main result reformulated, typos corrected)

References in corpus (13)

• Tunneling and Metastability of continuous time Markov chains
• Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model
• Dirichlet's and Thomson's principles for non-selfadjoint elliptic operators with application to non-reversible metastable diffusion processes
• Metastable states, quasi-stationary distributions and soft measures
• Jump Markov models and transition state theory: the Quasi-Stationary Distribution approach
• On the noise-induced passage through an unstable periodic orbit II: General case
• From random Poincaré maps to stochastic mixed-mode-oscillation patterns
• The Poincare map of randomly perturbed periodic motion
• Metastable Markov chains: from the convergence of the trace to the convergence of the finite-dimensional distributions
• Sharp spectral asymptotics for non-reversible metastable diffusion processes
• Metastable distributions of Markov chains with rare transitions
• Spectral theory for random Poincaré maps
• Analysis of metastable behavior via solutions of Poisson equations

Cited by in corpus (1)

• Metastability in the stochastic nearest-neighbor Kuramoto model of coupled phase oscillators