Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions

Kipnis, Claude; Varadhan, S. R. S.

doi:10.1007/bf01210789

Cited by 680 publications

(800 citation statements)

References 3 publications

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“…This assumption, which will only be needed when n − k = 1, will be explained later in the context of Theorems 2 and 3. It is not needed for n − k ≥ 2, when Z has finite variance and Theorem 1 is for the most part a consequence of well known limit theorems, in particular the central limit theorem for reversible Markov chains as formulated by Kipnis and Varadhan in [20]. …”

Section: Diffusivity Spectrum and Mean Exit Timementioning

confidence: 99%

“…(For an early study of two parallel plates case, see in [6].) Infinite variance of the in-between collisions displacements requires a generalization of the central limit theorem of Kipnis and Varadhan in [20] that is proved in this paper. Now, from an appropriate central limit theorem we obtain for τ the asymptotic expression where η only depends on the scattering characteristics at the microscopic scale determined by P .…”

Section: An Idealized Experiments and The Main Questionmentioning

confidence: 99%

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Diffusivity in multiple scattering systems

Chumley¹,

Feres²,

Zhang³

2015

Trans. Amer. Math. Soc.

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We consider random flights of point particles inside n-dimensional channels of the form R k ×B n−k , where B n−k is a ball of radius r in dimension n−k. The particle velocities immediately after each collision with the boundary of the channel comprise a Markov chain with a transition probabilities operator P that is determined by a choice of (billiard-like) random mechanical model of the particle-surface interaction at the "microscopic" scale. Our central concern is the relationship between the scattering properties encoded in P and the constant of diffusivity of a Brownian motion obtained by an appropriate limit of the random flight in the channel. Markov operators obtained in this way are natural (definition below), which means, in particular, that (1) the (at the surface) Maxwell-Boltzmann velocity distribution with a given surface temperature, when the surface model contains moving parts, or (2) the so-called Knudsen cosine law, when this model is purely geometric, is the stationary distribution of P . We show by a suitable generalization of a central limit theorem of Kipnis and Varadhan how the diffusivity is expressed in terms of the spectrum of P and compute, in the case of 2-dimensional channels, the exact values of the diffusivity for a class of parametric microscopic surface models of the above geometric type (2).

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Section: Diffusivity Spectrum and Mean Exit Timementioning

confidence: 99%

Section: An Idealized Experiments and The Main Questionmentioning

confidence: 99%

Diffusivity in multiple scattering systems

Chumley¹,

Feres²,

Zhang³

2015

Trans. Amer. Math. Soc.

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“…Indeed, an argument based on the central limit theorem [3,Ch. 3], [16] implies that at long times the particle performs an effective Brownian motion which is a Gaussian process, and hence the first two moments are sufficient to determine the process uniquely. The main goal of all theoretical investigations of noisy, non-equilibrium particle transport is the calculation of (1) and (2).…”

Section: Introductionmentioning

confidence: 99%

A multiscale approach to Brownian motors

Pavliotis

2005

Physics Letters A

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The problem of Brownian motion in a periodic potential, under the influence of external forcing, which is either random or periodic in time, is studied in this paper. Multiscale techniques are used to derive general formulae for the steady state particle current and the effective diffusion tensor. These formulae are then applied to calculate the effective diffusion coefficient for a Brownian particle in a periodic potential driven simultaneously by additive Gaussian white and colored noise. Our theoretical findings are supported by numerical simulations.

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“…Finally, we note that the sesquilinear form associated with the operator J can be written as (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) (…”

mentioning

confidence: 99%

“…On the other hand, Theorem 2.3(b) is very strong. It even implies results about the numerical range (and hence the spectrum) of operators F(J) for certain analytic functions F [10]; one case of interest is F(J) = J-1, which arises in the central limit theorem (see, e.g., [11,12]). …”

mentioning

confidence: 99%

Bounds on the 𝐿² spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality

Lawler

Sokal²

1988

Trans. Amer. Math. Soc.

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ABSTRACT. We prove a general version of Cheeger's inequality for discretetime Markov chains and continuous-time Markovian jump processes, both reversible and nonreversible, with general state space. We also prove a version of Cheeger's inequality for Markov chains and processes with killing. As an application, we prove L2 exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs.

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Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions

Cited by 680 publications

References 3 publications

Diffusivity in multiple scattering systems

Diffusivity in multiple scattering systems

A multiscale approach to Brownian motors

Bounds on the 𝐿² spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality

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