Short time large deviations of the KPZ equation

Abstract

We establish the Freidlin--Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation with multiplicative noise in one spatial dimension. That is, we introduce a small parameter $\sqrt{\epsilon }$ to the noise, and establish an LDP for the trajectory of the solution. Such a Freidlin--Wentzell LDP gives the short-time, one-point LDP for the KPZ equation in terms of a variational problem. Analyzing this variational problem under the narrow wedge initial data, we prove a quadratic law for the near-center tail and a $\frac{5}{2}$ law for the deep lower tail. These power laws confirm existing physics predictions Kolokolov and Korshunov (2007), Kolokolov and Korshunov (2009), Meerson, Katzav, and Vilenkin (2016), Le Doussal, Majumdar, Rosso, and Schehr (2016), and Kamenev, Meerson, and Sasorov (2016).

24 pages, 3 figures

References in corpus (35)

• Particles and fields in fluid turbulence
• Macroscopic fluctuation theory
• A theory of regularity structures
• Exact Large Deviation Function in the Asymmetric Exclusion Process
• Rare Events Statistics in Reaction--Diffusion Systems
• The one-dimensional KPZ equation and its universality class
• Large deviations for infinite dimensional stochastic dynamical systems
• The instanton method and its numerical implementation in fluid mechanics
• Large Deviations of Surface Height in the Kardar-Parisi-Zhang Equation
• Exact short-time height distribution in 1D KPZ equation and edge fermions at high temperature
• Large deviations for the height in 1D Kardar-Parisi-Zhang growth at late times
• Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions
• Optimal fluctuation approach to a directed polymer in a random medium
• Large deviations of surface height in the $1+1$-dimensional Kardar-Parisi-Zhang equation: exact long-time results for $\lambda H<0$
• Explicit solution of the optimal fluctuation problem for an elastic string in random potential
• Exact short-time height distribution in 1D KPZ equation with Brownian initial condition
• High-precision simulation of the height distribution for the KPZ equation
• Coulomb-gas electrostatics controls large fluctuations of the KPZ equation
• Short-time height distribution in 1d KPZ equation: starting from a parabola
• Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions
• Lower tail of the KPZ equation
• Soliton approach to the noisy Burgers equation: Steepest descent method
• Linear statistics and pushed Coulomb gas at the edge of beta random matrices: four paths to large deviations
• Height distribution tails in the Kardar-Parisi-Zhang equation with Brownian initial conditions
• Negative velocity fluctuations of pulled reaction fronts
• Simple derivation of the $(-\lambda H{)}^{5/2}$ tail for the 1D KPZ equation
• Large fluctuations of the KPZ equation in a half-space
• Optimal paths of non-equilibrium stochastic fields: the Kardar-Parisi-Zhang interface as a test case
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line
• A Riemann-Hilbert approach to the lower tail of the KPZ equation
• Asymptotics of the density of parabolic Anderson random fields
• Some recent progress in singular stochastic PDEs
• Exact lower tail large deviations of the KPZ equation
• Regularity and strict positivity of densities for the nonlinear stochastic heat equation
• Large deviations for the KPZ equation from the KP equation

Cited by in corpus (12)

• The inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation
• Inverse scattering solution of the weak noise theory of the Kardar-Parisi-Zhang equation with flat and Brownian initial conditions
• Observing symmetry-broken optimal paths of stationary Kardar-Parisi-Zhang interface via a large-deviation sampling of directed polymers in random media
• KPZ equation with a small noise, deep upper tail and limit shape
• Integrability in the weak noise theory
• KPZ fluctuations in finite volume
• Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition
• Short- and long-time path tightness of the continuum directed random polymer
• High moments of the SHE in the clustering regimes
• Upper-tail large deviation principle for the ASEP
• Inverse Scattering Method Solves the Problem of Full Statistics of Nonstationary Heat Transfer in the Kipnis-Marchioro-Presutti Model
• A nonlinear Strassen law for singular SPDEs