Coupling and Hydrodynamic Limit for the Inclusion Process

Abstract

We show propagation of local equilibrium for the symmetric inclusion process (SIP) after diffusive rescaling of space and time, as well as the local equilibrium property of the non-equilibrium steady state in the boundary driven SIP. The main tool is self-duality and a coupling between n SIP particles and n independent random walkers.

Condensation of SIP particles and sticky Brownian motion

“…Notice that once fixed the initial position of particles, the particles keep the same label. This process was studied for the first time in [16] and later on [3], but in contrast to [3], we do not restrict ourselves to the nearest neighbor case, hence at any Poisson clock ring the value of w(t) can change by r units, with r ∈…”

“…x j j (t)) = (B sbm,x i −x j (t), Bx i +x j (2t−τ (t))) (157) Here B sbm,x i −x j (t) is a sticky Brownian motion with stickiness parameter √ 2γ, and diffusion constant χ, started from x i − x j and where τ (t) is the corresponding local time-change defined in (16), and Bx i +x j (2t − τ (t)) is another brownian motion and diffusion constant χ, independent from B sbm (t) started from x i + x j .…”

Condensation of SIP Particles and Sticky Brownian Motion

“…Theorem 3.1 Let {η α(N ,t) : t ≥ 0} be the time-rescaled inclusion process, with infinistesimal generator (11), in configuration space. Consider the fluctuation field X N (η, ϕ, t) given by (14).…”

“…Notice that the labels of the particles are fixed at time zero and do not vary thereafter. This process was studied for the first time in [11] and later on [1], but in contrast to [1], we do not restrict ourselves to the nearest-neighbor case, hence any time a particle moves the value of w(t) can change by r units, with r ∈…”

Quantitative ergodicity for the symmetric exclusion process with stationary initial data

“…2.5.2], or the special class of reversible stochastic lattice gases in [17, § 4.1]. Another simple model for which the quantitative ergodicity could be investigated is the inclusion process (SIP), indeed this model is self-dual and a coupling with independent random walks has been constructed in [21]. For the generic case of reversible stochastic lattice gases where the invariant measure is not a product measure, it seems however difficult that coupling arguments suffice to establish the quantitative ergodicity, cfr.…”