[Paper Review] Ergodicity for infinite particle systems with locally conserved quantities

J. Inglis, Mikhail Neklyudov+1 more|arXiv (Cornell University)|Feb 1, 2010

Stochastic processes and statistical mechanics14 references3 citations

TL;DR

This paper establishes ergodicity for infinite-dimensional Markov processes with locally conserved quantities, using Hörmander-type generators that lack spectral gaps. Despite degeneracy and absence of Hörmander's condition, the system exhibits polynomial convergence to equilibrium, and a Liggett-Nash inequality holds, demonstrating long-time relaxation without exponential speed.

ABSTRACT

We analyse certain degenerate infinite dimensional sub-elliptic generators, and obtain estimates on the long-time behaviour of the corresponding Markov semigroups that describe a certain model of heat conduction. In particular, we establish ergodicity of the system for a family of invariant measures, and show that the optimal rate of convergence to equilibrium is polynomial. Consequently, there is no spectral gap, but a Liggett-Nash type inequality is shown to hold.

Motivation & Objective

• To analyze degenerate infinite-dimensional sub-elliptic generators arising in stochastic dynamics with locally conserved quantities.
• To establish ergodicity of the Markov semigroup despite the absence of a spectral gap due to local conservation.
• To characterize the long-time behavior of the system, particularly the rate of convergence to equilibrium.
• To prove the validity of a Liggett-Nash-type inequality in the absence of spectral gap, providing a functional analytic substitute.
• To demonstrate that finite-dimensional approximations fail to capture ergodicity, necessitating direct infinite-dimensional analysis.

Proposed method

• The authors use a Hörmander-type generator of the form $\mathcal{L} = \sum_{\gamma} \mathbf{X}_{\gamma}^2$, where vector fields $\mathbf{X}_{\gamma}$ are homogeneous under a dilation generator $D$.
• They analyze the Lie algebra generated by the vector fields to show that degeneracy is not removed by commutators, even to infinite order.
• A key technique involves deriving a differential inequality for $ \frac{d}{ds} P_{t-s} |\nabla f_s|^2 $, leading to a bound on the gradient norm.
• By introducing a damping term $-\beta D$, the authors analyze phase transitions in convergence behavior, distinguishing between exponential and algebraic decay.
• The proof relies on Sobolev-type symmetry and weak/strong continuity of the semigroup in $L^p$-spaces.
• Homogenization and functional central limit theorems are applied to confirm the asymptotic behavior of additive functionals under the invariant measure.

Experimental results

Research questions

• RQ1 Can ergodicity be established for infinite particle systems with locally conserved quantities when the generator lacks a spectral gap?
• RQ2 What is the optimal rate of convergence to equilibrium in such systems, and can it be characterized without spectral gap assumptions?
• RQ3 How does the presence of a damping term $-\beta D$ affect the convergence dynamics and induce a phase transition?
• RQ4 To what extent do finite-dimensional approximations fail to capture the ergodic properties of the infinite system?
• RQ5 Does a Liggett-Nash-type inequality hold in this degenerate, non-Hörmander setting, and what does it imply for long-time behavior?

Key findings

• The Markov semigroup generated by $\mathcal{L}$ is ergodic for a family of invariant measures $\mu_{r\mathbf{G}} \propto e^{-V/r} dx$, despite the absence of a spectral gap.
• The optimal rate of convergence to equilibrium is polynomial, not exponential, as shown by the inequality $ |\nabla f_t|^2 \leq e^{-2(\beta - \eta)t} P_t |\nabla f|^2 $.
• For $\beta > \beta_c$, the system exhibits exponential decay to equilibrium; for $\beta < \beta_c$, convergence is algebraic, indicating a phase transition.
• A Liggett-Nash-type inequality holds, providing a functional analytic substitute for the spectral gap in the absence of strong non-degeneracy.
• The infinite-dimensional system is ergodic even though finite-dimensional truncations are not, due to the formal divergence of the conserved quantity $V$.
• A functional central limit theorem holds for additive functionals under the invariant measure, confirming diffusive scaling in the long-time limit.