Abstract
We prove the convergence of the Sasamoto-Spohn model in equilibrium to the energy solution of the stochastic Burgers equation on the whole line. The proof, which relies on the second order Boltzmann-Gibbs principle, follows the approach of [9] and does not use any spectral gap argument.
Highlights
We prove the convergence of the Sasamoto-Spohn model in equilibrium to the energy solution of the stochastic Burgers equation on the whole line
The goal of this note is to show the convergence of a certain discretization of the stochastic Burgers equation:
It appears as the scaling limit of a wide range of particle systems [4, 8], directed polymer models [1, 20] and interacting diffusions [6], and constitutes a central element in a vast family of models known as the KPZ universality class [5, 21]
Summary
The goal of this note is to show the convergence of a certain discretization of the stochastic Burgers equation:. While the discretization of the second derivative and noise are quite straightforward, there are a priori several ways to discretize the nonlinearity in Burgers equation. This particular choice is motivated by two reasons: first, it only involves nearest neighbor sites and, second, it yields the explicit invariant measure μ = ρ⊗Z, where dρ(x) = √1 e−x2/2dx. Our result states the convergence of the discrete equations (1.2) to Burgers equation in the sense of energy solutions (see Section 2 for a precise definition). √ n ψj, j∈Z respectively, for φ ∈ L2(R) and ψ ∈ l2(Z)
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