2024/12/18 by Seonwoo Kim, Claudio Landim, Kim, Seonwoo +1
Mathematics · #60F10 #60J27 #60J45 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2412.13515
openalex publication_date 2024/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a sequence of continuous-time Markov chains (X(n)t:t≥ 0) evolving on a fixed finite state space V. Let In be the measure-current large deviations rate functional for X(n)t, as t→∞. Under a hypothesis on the jump rates, we prove that In can be written as In = \mathbf I(0) + ∑1≤ p≤ \mathfrak q (1/θ(p)n) \mathbf I(p) for some rate functionals \mathbf I(p). The weights θ(p)n correspond to the time-scales at which the sequence of Markov chains X(n)t evolves among the metastable wells, and the rate functionals \mathbf I(p) characterise the asymptotic Markovian dynamics among these wells. This expansion provides therefore an alternative description of the metastable behavior of a sequence of Markovian dynamics. Together with the results in \citebgl-24,l-gamma, this work finishes the project of characterising the hierarchical metastable behavior of finite-state Markov chains by means of the Γ-expansion of large deviations rate functionals. In addition, we present optimal conditions under which the measure (Donsker-Varadhan) or the measure-current large deviations rate functional determines the original dynamics, and calculate the first and second derivatives of the measure large deviations rate functional, thereby generalising the results for i.i.d. random variables.