Full $\Gamma$-expansion of reversible Markov chains level two large deviations rate functionals
Abstract
Let $\Xi_n \subset \mathbb R^d$, $n\ge 1$, be a sequence of finite sets and consider a $\Xi_n$-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 $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/\theta^{(p)}_n)\, I^{(p)}$, where $I^{(p)}\colon \mathscr P(\mathbb R^d) \to [0,+\infty]$ represent rate functionals independent of $n$ and $\theta^{(p)}_n$ sequences such that $\theta^{(1)}_n \to\infty$, $\theta^{(p)}_n / \theta^{(p+1)}_n \to 0$ for $1\le p< q$. The speed $\theta^{(p)}_n$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2023
- DOI:
- 10.48550/arXiv.2303.00671
- arXiv:
- arXiv:2303.00671
- Bibcode:
- 2023arXiv230300671L
- Keywords:
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- Mathematics - Probability;
- Condensed Matter - Statistical Mechanics;
- Mathematics - Analysis of PDEs;
- 60F10;
- 60J27;
- 60J45;
- 60K35
- E-Print:
- 39 pages. More explanations added in various parts of the text + introduction extended. Journal version