KPZ reloaded

Abstract

We analyze the one-dimensional periodic Kardar-Parisi-Zhang equation in the language of paracontrolled distributions, giving an alternative viewpoint on the seminal results of Hairer. Apart from deriving a basic existence and uniqueness result for paracontrolled solutions to the KPZ equation we perform a thorough study of some related problems. We rigorously prove the links between KPZ equation, stochastic Burgers equation, and (linear) stochastic heat equation and also the existence of solutions starting from quite irregular initial conditions. We also build a partial link between energy solutions as introduced by Goncalves and Jara and paracontrolled solutions. Interpreting the KPZ equation as the value function of an optimal control problem, we give a pathwise proof for the global existence of solutions and thus for the strict positivity of solutions to the stochastic heat equation. Moreover, we study Sasamoto-Spohn type discretizations of the stochastic Burgers equation and show that their limit solves the continuous Burgers equation possibly with an additional linear transport term. As an application, we give a proof of the invariance of the white noise for the stochastic Burgers equation which does not rely on the Cole-Hopf transform.

111 pages

References in corpus (7)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• The one-dimensional KPZ equation and its universality class
• KPZ equation limit of higher-spin exclusion processes
• Rough differential equations driven by signals in Besov spaces
• A central limit theorem for the KPZ equation
• Approximating three-dimensional Navier-Stokes equations driven by space-time white noise
• On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations

Cited by in corpus (36)

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• Global solutions to elliptic and parabolic ${\Phi }^4$ models in Euclidean space
• Lower tail of the KPZ equation
• The Kardar-Parisi-Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions
• Log-Sobolev inequality for the continuum sine-Gordon model
• Multidimensional SDE with distributional drift and Lévy noise
• Discretisation of regularity structures
• Rough differential equations driven by signals in Besov spaces
• Stationary solutions to the stochastic Burgers equation on the line
• Algebraic renormalisation of regularity structures
• How flat is flat in random interface growth?
• On global solutions of the random Hamilton-Jacobi equations and the KPZ problem
• Yang-Mills measure on the two-dimensional torus as a random distribution
• Field-Theoretic Thermodynamic Uncertainty Relation -- General formulation exemplified with the Kardar-Parisi-Zhang equation
• A diagram-free approach to the stochastic estimates in regularity structures
• KPZ equation limit of stochastic higher spin six vertex model
• Riemann surfaces for KPZ with periodic boundaries
• Lyapunov exponents of the SHE for general initial data
• Quantitative heat kernel estimates for diffusions with distributional drift
• Weak coupling limit of the Anisotropic KPZ equation
• KPZ equation limit of sticky Brownian motion
• Fluctuation Exponents of the KPZ equation on a large torus
• KPZ fluctuations in finite volume
• High temperature behaviors of the directed polymer on a cylinder
• Local KPZ behavior under arbitrary scaling limits
• Approximating three-dimensional magnetohydrodynamics system forced by space-time white noise
• The KPZ equation and moments of random matrices
• Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential
• Jointly invariant measures for the Kardar-Parisi-Zhang Equation
• Multiplicative SHE limit of random walks in space-time random environments
• An Additive-Noise Approximation to Keller-Segel-Dean-Kawasaki Dynamics: Local Well-Posedness of Paracontrolled Solutions
• Ergodicity of 2D singular stochastic Navier-Stokes equations
• Temporal increments of the KPZ equation with general initial data
• Characterization of Gradient Condition for Asymmetric Partial Exclusion Processes and Their Scaling Limits