Convergence to the stochastic Burgers equation from a degenerate microscopic dynamics

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 [7]. 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. 1 arXiv:1603.08975v3 [math.PR]

Energy solutions of KPZ are unique

“…Following [GJ14] it has been shown for a variety of models that their fluctuations subsequentially converge to energy solutions of the KPZ equation or the SBE, for example for zero range processes and kinetically constrained exclusion processes in [GJS15], various exclusion processes in [GS15,FGS16,BGS16,GJ16], interacting Brownian motions in [DGP16], and Hairer-Quastel type SPDEs in [GP16]. This is coherent with the conjecture that the SBE/KPZ equation describes the universal behavior of a wide class of conservative dynamics or interface growth models in the particular limit where the asymmetry is "small" (depending on the spatial scale), the so called weak KPZ universality conjecture, see [Cor12,Qua14,QS15,Spo16].…”

Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP

“…Collecting all the previous computations we get that (4.13) is bounded from above by C(T ) m 1 m 4−2k , which is finite as long as 2k − 4 > 1. Now we prove (2). For that purpose, at first we notice that from the previous computations we have: for k > 5 from which (4.18) follows.…”

“…Later in [19] it is assumed that the models satisfy a spectral gap bound which does not need to be uniform in the density of particles and allows more general jump rates. More recently in [11,20], and then in [2], a new proof of the BGP has permitted to extend the previous results to models which do not need to satisfy a spectral gap bound, as, for example, symmetric exclusion processes with a slow bond, and microscopic dynamics that allow degenerate exchange rates. In this paper, we adapt that strategy, which turns out to be quite robust, to our finite model with stochastic boundary reservoirs.…”

“…Note that the definition of the Ornstein-Uhlenbeck process given in Proposition 3.1 and of the energy solution to the stochastic Burgers equation (3.4) when E = 0, are equivalent. Indeed, the only part which is not obvious is that the Ornstein-Uhlenbeck process satisfies conditions (2) and (4) in Theorem 3.3. But both of these will follow from our convergence result Theorem 3.15 with E = 0.…”

“…Only recently Hairer has developed a meaningful notion of solution for the KPZ equation and proved existence and uniqueness of such solutions with periodic boundary conditions, see [26]. His approach uses rough path integrals to construct the nonlinearity (∇ (t, u)) 2 , and it has inspired the development of new technologies (regularity structures [27] and paracontrolled distributions [22]) for the so-called singular stochastic partial differential equations (SPDEs).…”