Nonlinear fluctuations of weakly asymmetric interacting particle systems

Abstract

We introduce what we call the second-order Boltzmann-Gibbs principle, which allows to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear function of the conserved quantity. This replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states. As an application of this second-order Boltzmann-Gibbs principle, we introduce the notion of energy solutions of the KPZ and stochastic Burgers equations. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, stationary, weakly asymmetric, conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic Burgers equation. We also show that the fluctuations of the height function associated to these models are given by energy solutions of the KPZ equation in this sense. Unfortunately, we lack a uniqueness result for these energy solutions. We conjecture these solutions to be unique, and we show some regularity results for energy solutions of the KPZ/Burgers equation, supporting this conjecture.

43 pages, to appear in Archive for Rational Mechanics and Analysis

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• A stochastic Burgers equation from a class of microscopic interactions
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• Energy solutions of KPZ are unique
• KPZ equation limit of stochastic higher spin six vertex model
• The Hairer--Quastel universality result in equilibrium
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• On microscopic derivation of a fractional stochastic Burgers equation
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• Some recent progress in singular stochastic PDEs
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• KPZ fluctuations in finite volume
• Hyperviscous stochastic Navier-Stokes equations with white noise invariant measure in two dimensions
• Local KPZ behavior under arbitrary scaling limits
• The KPZ equation and moments of random matrices
• On the scaling limits of weakly asymmetric bridges
• The Einstein relation for the KPZ equation
• Kardar-Parisi-Zhang Equation from Long-Range Exclusion Processes
• Hairer-Quastel universality in non-stationarity via energy solution theory
• Non-Stationary KPZ equation from ASEP with slow bonds
• From ABC to KPZ
• Multiplicative SHE limit of random walks in space-time random environments
• Fractal Geometry of the Valleys of the Parabolic Anderson Equation
• Stochastic Burgers Equation via Energy Solutions from Non-Stationary Particle Systems
• Quadratic Fluctuations of the Simple Exclusion Process
• Hydrodynamic limit for a boundary driven super-diffusive symmetric exclusion
• Temporal increments of the KPZ equation with general initial data
• Spectral gap of the symmetric inclusion process
• Characterization of Gradient Condition for Asymmetric Partial Exclusion Processes and Their Scaling Limits
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• Existence and Uniqueness of Energy Solutions to the Stochastic Diffusive Surface Quasi-Geostrophic Equation with Additive Noise
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