Scaling Limits of Additive Functionals of Interacting Particle Systems

Probab. Theory Relat. Fields

Sethuraman

2015

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ABSTRACT. With respect to a class of long-range exclusion processes on Z d , with single particle transition rates of order | · | −(d+α) , starting under Bernoulli invariant measure ν ρ with density ρ, we consider the fluctuation behavior of occupation times at a vertex and more general additive functionals. Part of our motivation is to investigate the dependence on α, d and ρ with respect to the variance of these functionals and associated scaling limits. In the case the rates are symmetric, among other results, we find the scaling limits exhaust a range of fractional Brownian motions with Hurst parameter H ∈ [1/2,3/4].However, in the asymmetric case, we study the asymptotics of the variances, which when d = 1 and ρ = 1/2 points to a curious dichotomy between long-range strength parameters 0 < α ≤ 3/2 and α > 3/2. In the former case, the order of the occupation time variance is the same as under the process with symmetrized transition rates, which are calculated exactly. In the latter situation, we provide consistent lower and upper bounds and other motivations that this variance order is the same as under the asymmetric shortrange model, which is connected to KPZ class scalings of the space-time bulk mass density fluctuations.

Section: Finite-range Models: Asymmetric Transitions and Kpz Exponentsmentioning

confidence: 99%

Occupation times of long-range exclusion and connections to KPZ class exponents

Bernardin

Probab. Theory Relat. Fields

Sethuraman

2015

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“…For these models, one should be able to repeat our multi-scale argument for higher degree polynomial functions, as done in [12]. These are works in progress.…”

Section: Introductionmentioning

confidence: 99%

Crossover to the Stochastic Burgers Equation for the WASEP with a Slow Bond

Franco¹,

Simon

2016

Commun. Math. Phys.

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ABSTRACT. We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter ρ ∈ (0,1). The rate of passage of particles to the right (resp. left) is 1 2 + a 2n γ (resp. 1 2 − a 2n γ ) except at the bond of vertices {−1,0} where the rate to the right (resp. left) is given bywe show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if γ > 1 2 , while for γ = 1 2 it is an energy solution of the stochastic Burgers equation. For γ ≥ β = 1, it is an Ornstein-Uhlenbeck process associated to the heat equation with Robin's boundary conditions. For γ ≥ β > 1, the limit density fluctuation field is an Ornstein-Uhlenbeck process associated to the heat equation with Neumann's boundary conditions.

“…The first statement is a consequence of the Energy Estimate (2.15); the proof of Theorem 2.1 in [16] can be readily adapted to our situation. Starting from (5.2), the proof of Theorem 2.5 in [16] can be adapted to prove the convergence stated in the second statement. The third statement is a consequence of this convergence combined with Theorem 2.14 in [3].…”

Section: Discussion and Remarksmentioning

confidence: 85%

Density fluctuations for exclusion processes with long jumps

Probab. Theory Relat. Fields

Jara

2017

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We show that the stationary density fluctuations of exclusion processes with long jumps, whose rates are of the form c ± | y − x| −(1+α) where c± depends on the sign of y − x, are given by a fractional Ornstein-Uhlenbeck process for α ∈ (0, 3 2 ). When α = 3 2 we show that the density fluctuations are tight, in a suitable topology, and that any limit point is an energy solution of the fractional Burgers equation, previously introduced in [18] in the finite volume setting.where t is a Brownian motion, ρ is the generator of an α-stable skewed Lévy process given in (2.20), ( ρ ) * is its adjoint and 1/2 is the symmetric part of ρ . In the case α = 3/2 we prove that the density fluctuation field is tight and any limit point is an energy solution of the fractional Burgers equation