This article considers a class of metastable non-reversible diffusion processes whose invariant measure is a Gibbs measure associated with a Morse potential. In a companion paper [32], we proved the Eyring-Kramers formula for the corresponding class of metastable diffusion processes. In this article, we further develop this result by proving that a suitably time-rescaled metastable diffusion process converges to a Markov chain on the deepest metastable valleys. This article is also an extension of [45], which considered the same problem for metastable reversible diffusion processes. Our proof is based on the recently developed resolvent approach to metastability.
In this article, we prove the Eyring-Kramers formula for non-reversible metastable diffusion processes that have a Gibbs invariant measure. Our result indicates that non-reversible processes exhibit faster metastable transitions between neighborhoods of local minima, compared to the reversible process considered in [Bovier, Eckhoff, Gayrard, and Klein, J. Eur. Math. Soc. 6: 399-424, 2004]. Therefore, by adding non-reversibility to the model, we can indeed accelerate the metastable transition. Our proof is based on the potential theoretic approach to metastability through accurate estimation of…
In this article, we perform quantitative analyses of metastable behavior of an interacting particle system known as the inclusion process. For inclusion processes, it is widely believed that the system nucleates the condensation of particles because of the attractive nature of the interaction mechanism. The metastable behavior of the inclusion processes corresponds to the movement of the condensate on a suitable time scale, and the computation of the corresponding time scale and the characterization of the scaling limit of the condensate motion are the main problems in the study of metastabili…
We consider translation-invariant interacting particle systems on the lattice with finite local state space admitting at least one Gibbs measure as a time-stationary measure. The dynamics can be irreversible but should satisfy some mild non-degeneracy conditions. We prove that weak limit points of any trajectory of translation-invariant measures, satisfying a non-nullness condition, are Gibbs states for the same specification as the time-stationary measure. This is done under the additional assumption that zero entropy loss of the limiting measure w.r.t. the time-stationary measure implies tha…
Consider a sequence of continuous-time Markov chains $(X^{(n)}_t:t\ge 0)$ evolving on a fixed finite state space $V$. Let $I_n$ be the level two large deviations rate functional for $X^{(n)}_t$, as $t\to\infty$. Under a hypothesis on the jump rates, we prove that $I_n$ can be written as $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/θ^{(p)}_n) \, I^{(p)}$ for some rate functionals $I^{(p)}$. The weights $θ^{(p)}_n$ correspond to the time-scales at which the sequence of Markov chains $X^{(n)}_t$ exhibit a metastable behavior, and the zero level sets of the rate functionals $I^{(p)}$ identify the me…
The rate of convergence of the Gibbs sampler for the generalized one-dimensional Ising model is determined by the second largest eigenvalue of its transition matrix in absolute value denoted by $β^*$. In this paper we generalize a result from Shiu and Chen $(2015)$ for the one-dimensional Ising model with two states which gives a bound for $β^*$ . The method is based on Diaconis and Stroock bound for reversible Markov processes. The new bound presented in this paper improves Ingrassia's $(1994)$ result.
We investigate the metastable behavior of reversible Markov chains on possibly countable infinite state spaces. Based on a new definition of metastable Markov processes, we compute precisely the mean transition time between metastable sets. Under additional size and regularity properties of metastable sets, we establish asymptotic sharp estimates on the Poincaré and logarithmic Sobolev constant. The main ingredient in the proof is a capacitary inequality along the lines of V. Maz'ya that relates regularity properties of harmonic functions and capacities. We exemplify the usefulness of this new…
Continuous time Feynman-Kac measures on path spaces are central in applied probability, partial differential equation theory, as well as in quantum physics. This article presents a new duality formula between normalized Feynman-Kac distribution and their mean field particle interpretations. Among others, this formula allows us to design a reversible particle Gibbs-Glauber sampler for continuous time Feynman-Kac integration on path spaces. This result extends the particle Gibbs samplers introduced by Andrieu-Doucet-Holenstein [2] in the context of discrete generation models to continuous time F…
The main contribution of the current study is two-fold. First, we investigate the energy landscape of the Ising and Potts models on finite two-dimensional lattices without external fields in the low temperature regime. The complete analysis of the energy landscape of these models was unknown because of its complicated plateau saddle structure between the ground states. We characterize this structure completely in terms of a random walk on the set of sub-trees of a ladder graph. Second, we provide a considerable simplification of the well-known potential-theoretic approach to metastability. In …
We consider the problem of metastability for stochastic reversible dynamics with exponentially small transition probabilities. We generalize previous results in several directions. We give an estimate of the spectral gap of the transition matrix and of the mixing time of the associated dynamics in terms of the maximal stability level. These model-independent results hold in particular for a large class of Probabilistic Cellular Automata (PCA), which we then focus on. We consider the PCA in a finite volume, at small and fixed magnetic field, and in the limit of vanishing temperature. This model…
The hitting and mixing times are two fundamental quantities associated with Markov chains. In Peres and Sousi[PS2015] and Oliveira[Oli2012], the authors show that the mixing times and "worst-case" hitting times of reversible Markov chains on finite state spaces are equal up to some universal multiplicative constant. We use tools from nonstandard analysis to extend this result to reversible Markov chains on general state spaces that satisfy the strong Feller property. Finally, we show that this asymptotic equivalence can be used to find bounds on the mixing times of a large class of Markov chai…
We examine two analytical characterisation of the metastable behavior of a Markov chain. The first one expressed in terms of its transition probabilities, and the second one in terms of its large deviations rate functional. Consider a sequence of continuous-time Markov chains $(X^{(n)}_t:t\ge 0)$ evolving on a fixed finite state space $V$. Under a hypothesis on the jump rates, we prove the existence of times-scales $θ^{(p)}_n$ and probability measures with disjoint supports $π^{(p)}_j$, $j\in S_p$, $1\le p \le q$, such that (a) $θ^{(1)}_n \to \infty$, $θ^{(k+1)}_n/θ^{(k)}_n \to \infty$, (b) fo…
