Full $Γ$-expansion of reversible Markov chains level two large deviations rate functionals
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918234778763264 |
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| author | Landim, Claudio Misturini, Ricardo Sau, Federico |
| author_facet | Landim, Claudio Misturini, Ricardo Sau, Federico |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_00671 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Full $Γ$-expansion of reversible Markov chains level two large deviations rate functionals Landim, Claudio Misturini, Ricardo Sau, Federico Probability Statistical Mechanics Analysis of PDEs 60F10, 60J27, 60J45, 60K35 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. |
| title | Full $Γ$-expansion of reversible Markov chains level two large deviations rate functionals |
| topic | Probability Statistical Mechanics Analysis of PDEs 60F10, 60J27, 60J45, 60K35 |
| url | https://arxiv.org/abs/2303.00671 |