[Paper Review] Sample path large deviations for a class of Markov chains related to disordered mean field models

Anton Bovier, Véronique Gayrard|ArXiv.org|May 5, 1999

Stochastic processes and statistical mechanics8 references3 citations

TL;DR

This paper establishes a sample path large deviation principle for a class of discrete-time Markov chains on a lattice with scaling parameter $\varepsilon$, where transition probabilities depend smoothly on position and time except near boundaries. The rate function is identified as an action functional derived from a time-dependent Lagrangian, providing a rigorous large deviation framework for mean field models with quenched disorder.

ABSTRACT

We prove a large deviation principle on path space for a class of discrete time Markov processes whose state space is the intersection of a regular domain $Ł\subset \R^d$ with some lattice of spacing $\e$. Transitions from $x$ to $y$ are allowed if $\e^{-1}(x-y)\in \D$ for some fixed set of vectors $\D$. The transition probabilities $p_\e(t,x,y)$, which themselves depend on $\e$, are allowed to depend on the starting point $x$ and the time $t$ in a sufficiently regular way, except near the boundaries, where some singular behaviour is allowed. The rate function is identified as an action functional which is given as the integral of a Lagrange function. %of time dependent relativistic classical mechanics. Markov processes of this type arise in the study of mean field dynamics of disordered mean field models.

Motivation & Objective

• To develop a large deviation principle for Markov chains on a lattice with small spacing $\varepsilon$, modeling disordered mean field systems.
• To characterize the exponential decay rate of rare path events in such systems.
• To identify the rate function as an action functional derived from a time-dependent Lagrangian.
• To allow for time- and space-dependent transition probabilities with controlled singularities near boundaries.
• To provide a rigorous probabilistic foundation for studying metastability and pathwise behavior in disordered mean field spin systems.

Proposed method

• Consider a Markov process on $\varepsilon \mathbb{Z}^d \cap \Lambda$, where $\Lambda \subset \mathbb{R}^d$ is a regular domain.
• Define transitions from $x$ to $y$ only if $\varepsilon^{-1}(x - y) \in \Delta$ for a fixed finite set $\Delta \subset \mathbb{Z}^d$.
• Allow transition probabilities $p_\varepsilon(t,x,y)$ to depend on time $t$ and position $x$ in a regular way, except near the boundary of $\Lambda$.
• Establish the large deviation principle on path space by proving exponential tightness and identifying the rate function via a variational formula.
• Identify the rate function as the integral of a Lagrangian over time, corresponding to a time-dependent relativistic classical mechanics action.
• Use weak convergence methods and compactness arguments to derive the large deviation principle under mild regularity conditions on the transition kernels.

Experimental results

Research questions

• RQ1 What is the large deviation rate function for sample paths of Markov chains with quenched disorder and spatial-temporal inhomogeneity on a lattice?
• RQ2 How does the rate function relate to a time-dependent Lagrangian in the context of mean field dynamics?
• RQ3 Can a large deviation principle be established for such chains despite boundary singularities in transition probabilities?
• RQ4 What is the asymptotic behavior of path probabilities as the lattice spacing $\varepsilon \to 0$?
• RQ5 How does the action functional emerge as the limit of discrete-time path probabilities?

Key findings

• The large deviation principle holds on the space of trajectories with a rate function given by the integral of a time-dependent Lagrangian over time.
• The rate function is identified as an action functional, which corresponds to the Hamiltonian of time-dependent relativistic classical mechanics.
• The convergence of the discrete-time Markov process to the continuous action functional is established under mild regularity conditions on the transition probabilities.
• Exponential tightness of the family of path measures is proven, ensuring the existence of the large deviation principle.
• The method applies even when transition probabilities exhibit singular behavior near the boundary of the domain $\Lambda$, under controlled conditions.
• The framework provides a rigorous pathwise description of rare events in disordered mean field spin systems, such as metastable transitions.