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A sufficient condition for the quasipotential to be the rate function of the invariant measure of countable-state mean-field interacting particle systems

Sarath Yasodharan, Rajesh Sundaresan

TL;DR

This paper investigates large deviations for the invariant measures of countable-state mean-field interacting particle systems and reveals that the Freidlin-Wentzell quasipotential $V$ is not universally the correct rate function. It identifies a barrier phenomenon where certain points in state space are unreachable by finite-horizon trajectories but are accessible in the stationary regime, causing $V$ to be infinite while a relative-entropy rate remains finite. The authors establish a constructive set of sufficient conditions, notably a $1/z$-decay in forward rates and related regularity, under which the stationary large deviations of the invariant measures are governed by $V$. They illustrate the necessity of these conditions with two explicit counterexamples: a system of non-interacting M/M/1 queues and a non-interacting WLAN system with constant forward rates, where $V$ diverges even though the relative-entropy rate remains finite. The work thus clarifies when the Freidlin-Wentzell quasipotential can be used as the rate function and suggests directions for extending uniform LDPs and alternative quasipotential formulations in non-locally compact, infinite-dimensional settings.

Abstract

This paper considers the family of invariant measures of Markovian mean-field interacting particle systems on a countably infinite state space and studies its large deviation asymptotics. The Freidlin-Wentzell quasipotential is the usual candidate rate function for the sequence of invariant measures indexed by the number of particles. The paper provides two counterexamples where the quasipotential is not the rate function. The quasipotential arises from finite horizon considerations. However there are certain barriers that cannot be surmounted easily in any finite time horizon, but these barriers can be crossed in the stationary regime. Consequently, the quasipotential is infinite at some points where the rate function is finite. After highlighting this phenomenon, the paper studies some sufficient conditions on a class of interacting particle systems under which one can continue to assert that the Freidlin-Wentzell quasipotential is indeed the rate function.

A sufficient condition for the quasipotential to be the rate function of the invariant measure of countable-state mean-field interacting particle systems

TL;DR

This paper investigates large deviations for the invariant measures of countable-state mean-field interacting particle systems and reveals that the Freidlin-Wentzell quasipotential is not universally the correct rate function. It identifies a barrier phenomenon where certain points in state space are unreachable by finite-horizon trajectories but are accessible in the stationary regime, causing to be infinite while a relative-entropy rate remains finite. The authors establish a constructive set of sufficient conditions, notably a -decay in forward rates and related regularity, under which the stationary large deviations of the invariant measures are governed by . They illustrate the necessity of these conditions with two explicit counterexamples: a system of non-interacting M/M/1 queues and a non-interacting WLAN system with constant forward rates, where diverges even though the relative-entropy rate remains finite. The work thus clarifies when the Freidlin-Wentzell quasipotential can be used as the rate function and suggests directions for extending uniform LDPs and alternative quasipotential formulations in non-locally compact, infinite-dimensional settings.

Abstract

This paper considers the family of invariant measures of Markovian mean-field interacting particle systems on a countably infinite state space and studies its large deviation asymptotics. The Freidlin-Wentzell quasipotential is the usual candidate rate function for the sequence of invariant measures indexed by the number of particles. The paper provides two counterexamples where the quasipotential is not the rate function. The quasipotential arises from finite horizon considerations. However there are certain barriers that cannot be surmounted easily in any finite time horizon, but these barriers can be crossed in the stationary regime. Consequently, the quasipotential is infinite at some points where the rate function is finite. After highlighting this phenomenon, the paper studies some sufficient conditions on a class of interacting particle systems under which one can continue to assert that the Freidlin-Wentzell quasipotential is indeed the rate function.

Paper Structure

This paper contains 30 sections, 25 theorems, 156 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Two counterexamples
  3. Non-interacting M/M/1 queues
  4. Non-interacting nodes in a wireless network
  5. Assumptions and main result
  6. Assumptions
  7. Main result
  8. Discussion and future directions
  9. Related literature
  10. Organisation
  11. Preliminaries
  12. Frequently used notation
  13. Notation related to the dynamics
  14. Process-level large deviations
  15. Invariant measure: Existence, uniqueness, and exponential tightness
  16. ...and 15 more sections

Key Result

Proposition 1.1

Assume assm:a1 and assm:a2. For each $N \geq 1$, $\mathscr{L}^N$ admits a unique invariant probability measure $\wp^N$. Further, the family $\{\wp^N, N \geq 1\}$ is exponentially tight in $\mathcal{M}_1(\mathcal{Z})$.

Figures (2)

  • Figure 1: Transition rates of an M/M/1 queue
  • Figure 2: Transition rates of a wireless node

Theorems & Definitions (48)

  • Proposition 1.1
  • Theorem 1.1
  • Definition 2.1: Large deviation principle
  • Definition 2.2: Uniform large deviation principle
  • Lemma 2.1
  • Theorem 2.1
  • Theorem 2.2: Non-variational representation; Léonard leonard-95-1
  • Remark 2.1
  • proof : Proof of Proposition \ref{['prop:invariant-measure-exp-tightness']}
  • Lemma 4.1
  • ...and 38 more