The study of equilibrium fluctuations of a tagged particle in finite-range simple exclusion processes has a long history. The belief is that the scaled centered tagged particle motion behaves as some sort of homogenized random walk. In fact, invariance principles have been proved in all dimensions d ≥ 1 when the single particle jump rate is unbiased, in d ≥ 3 when the jump rate is biased, and in d = 1 when the jump rate is in addition nearest-neighbor.The purpose of this article is to give some partial results in the open cases in d ≤ 2. Namely, we show the tagged particle motion is "diffusive" in the sense that upper and lower bounds are given for the tagged particle variance at time t on order O(t) in d = 2 when the jump rate is biased, and also in d = 1 when in addition the jump rate is not nearest-neighbor. Also, a characterization of the tagged particle variance is given. The main methods are in analyzing H −1 norm variational inequalities.
We consider a class of nearest-neighbor weakly asymmetric mass conservative
particle systems evolving on $\mathbb{Z}$, which includes zero-range and types
of exclusion processes, starting from a perturbation of a stationary state.
When the weak asymmetry is of order $O(n^{-\gamma})$ for $1/2<\gamma\leq1$, we
show that the scaling limit of the fluctuation field, as seen across process
characteristics, is a generalized Ornstein-Uhlenbeck process. However, at the
critical weak asymmetry when $\gamma=1/2$, we show that all limit points
satisfy a martingale formulation which may be interpreted in terms of a
stochastic Burgers equation derived from taking the gradient of the KPZ
equation. The proofs make use of a sharp "Boltzmann-Gibbs" estimate which
improves on earlier bounds.Comment: Published in at http://dx.doi.org/10.1214/13-AOP878 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Some growth asymptotics of a version of 'preferential attachment' random graphs are studied through an embedding into a continuous-time branching scheme. These results complement and extend previous work in the literature.
We apply the paracontrolled calculus to study the asymptotic behavior of a certain quasilinear PDE with smeared mild noise, which originally appears as the space-time scaling limit of a particle system in random environment on one dimensional discrete lattice. We establish the convergence result and show a local in time well-posedness of the limit stochastic PDE with spatial white noise. It turns out that our limit stochastic PDE does not require any renormalization. We also show a comparison theorem for the limit equation.
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