Abstract
In this paper we prove the convergence to the stochastic Burgers equation from one-dimensional interacting particle systems, whose dynamics allow the degeneracy of the jump rates. To this aim, we provide a new proof of the second order Boltzmann-Gibbs principle introduced in [Gon\c{c}alves, Jara 2014]. The main technical difficulty is that our models exhibit configurations that do not evolve under the dynamics - the blocked configurations - and are locally non-ergodic. Our proof does not impose any knowledge on the spectral gap for the microscopic models. Instead, it relies on the fact that, under the equilibrium measure, the probability to find a blocked configuration in a finite box is exponentially small in the size of the box. Then, a dynamical mechanism allows to exchange particles even when the jump rate for the direct exchange is zero.
Highlights
In the last few years there has been an intense research activity around the derivation of the stochastic Burgers equation (SBE), or its integrated counterpart, namely the Kardar-Parisi-Zhang (KPZ) equation, from one-dimensional weakly asymmetric conser-Convergence to the SBE from a degenerate dynamics vative interacting particle systems
The KPZ equation goes back to [16] where it has been proposed as the default stochastic partial differential equation (SPDE) ruling the evolution of the profile of a randomly growing interface
The KPZ equation is ill-posed, the problematic term being (∇h(t, x))2, which is not well-defined if h has the regularity we expect for a solution
Summary
In the last few years there has been an intense research activity around the derivation of the stochastic Burgers equation (SBE), or its integrated counterpart, namely the Kardar-Parisi-Zhang (KPZ) equation, from one-dimensional weakly asymmetric conser-. The key ingredient in the authors’ approach is the derivation of a second order Boltzmann-Gibbs principle (BGP2), which allows to make the non-linear term of the SBE equation emerge from the underlying microscopic dynamics This principle is obtained by a multiscale argument which consists in replacing a local function by another function whose support increases at each step and whose variance decreases in order to make the errors eventually vanish. In this article we give a step towards generalizing the previous results, since our main contribution is that we prove the convergence to the SBE from microscopic dynamics that allow degenerate exchange rates This goes towards showing that the more recent approach of [3, 10] to proving BGP2 is more flexible to situations which lack uniformity.
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