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Convergence of dynamical stationary fluctuations

Cyril Labbé, Benoît Laslier, Fabio Toninelli, Lorenzo Zambotti

Abstract

We present a general black box theorem that ensures convergence of a sequence of stationary Markov processes, provided a few assumptions are satisfied. This theorem relies on a control of the resolvents of the sequence of Markov processes, and on a suitable characterization of the resolvents of the limit. One major advantage of this approach is that it circumvents the use of the Boltzmann-Gibbs principle: for instance, we deduce in a rather simple way that the stationary fluctuations of the one-dimensional zero-range process converge to the stochastic heat equation. More importantly, it allows to establish results that were probably out of reach of existing methods: using the black box result, we are able to prove that the stationary fluctuations of a discrete model of ordered interfaces, that was considered previously in the statistical physics literature, converge to a system of reflected stochastic PDEs.

Convergence of dynamical stationary fluctuations

Abstract

We present a general black box theorem that ensures convergence of a sequence of stationary Markov processes, provided a few assumptions are satisfied. This theorem relies on a control of the resolvents of the sequence of Markov processes, and on a suitable characterization of the resolvents of the limit. One major advantage of this approach is that it circumvents the use of the Boltzmann-Gibbs principle: for instance, we deduce in a rather simple way that the stationary fluctuations of the one-dimensional zero-range process converge to the stochastic heat equation. More importantly, it allows to establish results that were probably out of reach of existing methods: using the black box result, we are able to prove that the stationary fluctuations of a discrete model of ordered interfaces, that was considered previously in the statistical physics literature, converge to a system of reflected stochastic PDEs.
Paper Structure (17 sections, 15 theorems, 169 equations, 3 figures)

This paper contains 17 sections, 15 theorems, 169 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A black box theorem
  3. Reduction to an identity on the resolvents
  4. Some technical tools
  5. Convergence of the resolvents
  6. The zero-range process
  7. Model and notations
  8. The SPDE
  9. Equicontinuity
  10. Discrete characterization and convergence
  11. Tightness
  12. Pair of reflected interfaces
  13. The SPDE
  14. Equicontinuity
  15. Discrete characterization and convergence
  16. ...and 2 more sections

Key Result

Theorem 2.6

Let $u_N$ start from its invariant measure $m_N$ for every $N\ge 1$. Given Assumptions Ass:Charac, Ass:State, Ass:Tight, Ass:Equi and Ass:IbPF, the sequence $(u_N)_N$ converges in law in $\mathbb{D}([0,\infty), E^{-})$ to the process $u$ starting from its invariant measure $m$.

Figures (3)

  • Figure 1: Example of a pair of reflected interfaces: $k=3$ and $k=9$ are contact points. At contact points, a reflection term prevents the jumps that would break the ordering to occur.
  • Figure 2: Dynamics at contact points
  • Figure 3: An example of allowed configuration for $k=3$ reflected interfaces. At positions $x_1,x_3$ there is a triple contact with a downward corner, and at $x_2$ a double contact with an upward corner. The rate at which $v_{(3)}$ flips upward at position $x_1$ (together with the other two interfaces) is $(2N)^2/6$, since $N_3(x_1)=2$. On the other hand, $v_{(2)}$ can also flip upward at $x_1$ (together with $v_{(1)}$) and it does so with rate $(2N)^2/4$.

Theorems & Definitions (32)

  • Theorem 2.6
  • Theorem 2.7: Modified Arzelà-Ascoli's theorem
  • Remark 2.8
  • Remark 2.9
  • proof
  • Lemma 2.10
  • proof
  • Lemma 2.11
  • proof
  • Lemma 3.1
  • ...and 22 more