We introduce what we call the second-order Boltzmann-Gibbs principle, which allows to replace local functionals of a conservative, onedimensional 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 applicat...

The short-time behavior of the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) growth equation with a flat initial condition is obtained from the exact expressions for the moments of the partition function of a directed polymer with one end point free and the other fixed. From these expressions, the short-time expansions of the lowest cumulants of the KPZ height field are exactly derived. The resul...

Growth in crystals can be { usually } described by field equations such as the Kardar-Parisi-Zhang (KPZ) equation. While crystalline structure characterized Euclidean geometry with its peculiar symmetries, growth dynamics creates a fractal at interface of crystal and medium, which turn determines growth. Recent work (The KPZ exponents for 2+ 1 dimensions, MS Gomes-Filho, ALA Penna, FA Oliveira;...

We establish the Freidlin–Wentzell Large Deviation Principle (LDP) for Stochastic Heat Equation with multiplicative noise in one spatial dimension. That is, we introduce a small parameter $$ \sqrt{\varepsilon } to noise, and an LDP trajectory of solution. Such gives short-time, one-point KPZ equation terms variational problem. Analyzing this problem under narrow wedge initial data, prove quadra...

The short-time evolution of a growing interface is studied analytically and numerically for the Kadar-Parisi-Zhang (KPZ) universality class. The scaling behavior of response and correlation functions is reminiscent of the “initial slip” behavior found in purely dissipative critical relaxation (model A). Unlike model A the initial slip exponent for the KPZ equation can be expressed by the dynami...

We study numerically the Kuramoto-Sivashinsky equation forced by external white noise in two space dimensions, that is a generic model for, e.g., surface kinetic roughening in the presence of morphological instabilities. Large scale simulations using a pseudospectral numerical scheme allow us to retrieve Kardar-Parisi-Zhang (KPZ) scaling as the asymptotic state of the system, as in the one-dime...

Non-linear field equations such as the KPZ equation for deposition and the Navier-Stokes equation for hydrodynamics are discussed by the derivation of transport equations for the correlation function of the field h (r, t), i.e. 〈h (r, t)h (r′, t ′)〉, where h satisfies a diffusion equation driven by a noise, f , defined as noise with a given spectrum, and containing a non linear term, Mhh, which...

We show that Fokker-Planck equation for chordal SLE process under a simple rescaling of the probability density can be traced to the minisuperspace Wheeler-de Witt equation for boundary operator in 2d Liouville gravity. Insertion of an operator, calculating SLE critical exponent, corresponds to adding matter contribution to WdW equation. This observation may be useful for understanding of why S...