74 research outputs found
Bulk diffusion in a system with site disorder
We consider a system of random walks in a random environment interacting via
exclusion. The model is reversible with respect to a family of disordered
Bernoulli measures. Assuming some weak mixing conditions, it is shown that,
under diffusive scaling, the system has a deterministic hydrodynamic limit
which holds for almost every realization of the environment. The limit is a
nonlinear diffusion equation with diffusion coefficient given by a variational
formula. The model is nongradient and the method used is the ``long jump''
variation of the standard nongradient method, which is a type of
renormalization. The proof is valid in all dimensions.Comment: Published at http://dx.doi.org/10.1214/009117906000000322 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Crossover distributions at the edge of the rarefaction fan
We consider the weakly asymmetric limit of simple exclusion process with
drift to the left, starting from step Bernoulli initial data with
so that macroscopically one has a rarefaction fan. We study the
fluctuations of the process observed along slopes in the fan, which are given
by the Hopf--Cole solution of the Kardar-Parisi-Zhang (KPZ) equation, with
appropriate initial data. For slopes strictly inside the fan, the initial data
is a Dirac delta function and the one point distribution functions have been
computed in [Comm. Pure Appl. Math. 64 (2011) 466-537] and [Nuclear Phys. B 834
(2010) 523-542]. At the edge of the rarefaction fan, the initial data is
one-sided Brownian. We obtain a new family of crossover distributions giving
the exact one-point distributions of this process, which converge, as
to those of the Airy
process. As an application, we prove moment and large deviation estimates for
the equilibrium Hopf-Cole solution of KPZ. These bounds rely on the apparently
new observation that the FKG inequality holds for the stochastic heat equation.
Finally, via a Feynman-Kac path integral, the KPZ equation also governs the
free energy of the continuum directed polymer, and thus our formula may also be
interpreted in those terms.Comment: Published in at http://dx.doi.org/10.1214/11-AOP725 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org). arXiv admin note: text overlap with
arXiv:1003.044
KPZ equation, its renormalization and invariant measures
The Kardar-Parisi-Zhang (KPZ) equation is a stochastic partial differential
equation which is ill-posed because the nonlinearity is marginally defined with
respect to the roughness of the forcing noise. However, its Cole-Hopf solution,
defined as the logarithm of the solution of the linear stochastic heat equation
(SHE) with a multiplicative noise, is a mathematically well-defined object. In
fact, Hairer [13] has recently proved that the solution of SHE can actually be
derived through the Cole-Hopf transform of the solution of the KPZ equation
with a suitable renormalization under periodic boundary conditions. This
transformation is unfortunately not well adapted to studying the invariant
measures of these Markov processes.
The present paper introduces a different type of regularization for the KPZ
equation on the whole line or under periodic boundary conditions,
which is appropriate from the viewpoint of studying the invariant measures. The
Cole-Hopf transform applied to this equation leads to an SHE with a smeared
noise having an extra complicated nonlinear term. Under time average and in the
stationary regime, it is shown that this term can be replaced by a simple
linear term, so that the limit equation is the linear SHE with an extra linear
term with coefficient 1/24. The methods are essentially stochastic analytic:
The Wiener-It\^o expansion and a similar method for establishing the
Boltzmann-Gibbs principle are used. As a result, it is shown that the
distribution of a two-sided geometric Brownian motion with a height shift given
by Lebesgue measure is invariant under the evolution determined by the SHE on
- β¦