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Full $Γ$-expansion of reversible Markov chains level two large deviations rate functionals

Claudio Landim, Ricardo Misturini, Federico Sau

TL;DR

This work develops a comprehensive multiscale Gamma-expansion for the level-two large deviations functional of reversible Markov chains on growing finite state spaces. It introduces a metastable framework with nested wells and multiple time-scales, and proves that the large deviations functional I_n decomposes as I_n = I^(0) + sum_p (1/θ_n^(p)) I^(p), where each I^(p) is the level-two LD functional of a reduced Markov chain describing the dynamics at the p-th time-scale. The analysis hinges on capacity methods, trace-process reductions, and Gamma-convergence, and is illustrated through random walks in potential fields, yielding explicit expressions for the limit functionals at each scale. The results extend previous metastability analyses to growing state spaces and provide a rigorous hierarchical description of metastable states and transitions across multiple speeds. The methodology offers a robust toolkit for understanding complex multiscale stochastic systems in discrete settings.

Abstract

Let $Ξ_n \subset \mathbb R^d$, $n\ge 1$, be a sequence of finite sets and consider a $Ξ_n$-valued, irreducible, reversible, continuous-time Markov chain $(X^{(n)}_t:t\ge 0)$. Denote by $\mathscr P(\mathbb R^d) $ the set of probability measures on $\mathbb R^d$ and by $I_n\colon \mathscr P(\mathbb R^d) \to [0,+\infty)$ the level two large deviations rate functional for $X^{(n)}_t$ as $t\to\infty$. We present a general method, based on tools used to prove the metastable behaviour of Markov chains, to derive a full expansion of $I_n$ expressing it as $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/θ^{(p)}_n)\, I^{(p)}$, where $I^{(p)}\colon \mathscr P(\mathbb R^d) \to [0,+\infty]$ represent rate functionals independent of $n$ and $θ^{(p)}_n$ sequences such that $θ^{(1)}_n \to\infty$, $θ^{(p)}_n / θ^{(p+1)}_n \to 0$ for $1\le p< q$. The speed $θ^{(p)}_n$ corresponds to the time-scale at which the Markov chains $X^{(n)}_t$ exhibits a metastable behavior, and the $I^{(p-1)}$ zero-level sets to the metastable states. To illustrate the theory we apply the method to random walks in potential fields.

Full $Γ$-expansion of reversible Markov chains level two large deviations rate functionals

TL;DR

This work develops a comprehensive multiscale Gamma-expansion for the level-two large deviations functional of reversible Markov chains on growing finite state spaces. It introduces a metastable framework with nested wells and multiple time-scales, and proves that the large deviations functional I_n decomposes as I_n = I^(0) + sum_p (1/θ_n^(p)) I^(p), where each I^(p) is the level-two LD functional of a reduced Markov chain describing the dynamics at the p-th time-scale. The analysis hinges on capacity methods, trace-process reductions, and Gamma-convergence, and is illustrated through random walks in potential fields, yielding explicit expressions for the limit functionals at each scale. The results extend previous metastability analyses to growing state spaces and provide a rigorous hierarchical description of metastable states and transitions across multiple speeds. The methodology offers a robust toolkit for understanding complex multiscale stochastic systems in discrete settings.

Abstract

Let , , be a sequence of finite sets and consider a -valued, irreducible, reversible, continuous-time Markov chain . Denote by the set of probability measures on and by the level two large deviations rate functional for as . We present a general method, based on tools used to prove the metastable behaviour of Markov chains, to derive a full expansion of expressing it as , where represent rate functionals independent of and sequences such that , for . The speed corresponds to the time-scale at which the Markov chains exhibits a metastable behavior, and the zero-level sets to the metastable states. To illustrate the theory we apply the method to random walks in potential fields.
Paper Structure (15 sections, 27 theorems, 158 equations)

This paper contains 15 sections, 27 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Setting
  4. Hypotheses
  5. Main results
  6. $\Gamma$-limsup of the trace
  7. Rate functional estimates
  8. Proof of Theorems \ref{['mt1']} and \ref{['mt2']}
  9. Random walks in potential fields
  10. Metastable behaviour of $X^{(n)}_t$
  11. $\Gamma$-expansion of ${\mathscr I}_n$
  12. $\Gamma$-convergence of ${\mathscr I}_n$
  13. $\Gamma$-convergence of $n\, {\mathscr I}_n$
  14. $\Gamma$-convergence of $n\, e^{d_pn}\, {\mathscr I}_n$
  15. Proof of Theorem \ref{['mt3']}

Key Result

Theorem 2.7

Assume that conditions (H0) -- (H5) are in force. Then, a full $\Gamma$-expansion of ${\mathscr I}_n$ as in Definition def1 is given by the speeds $(\theta^{(p)}_n, n\ge 1)$, $1\le p\le {\mathfrak q}$, and functionals ${\mathscr I}^{(p)}\colon {\mathscr P}(\Xi) \to [0, +\infty]$, $0\le p\le {\mathfr

Theorems & Definitions (63)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 53 more