Exact parabolic asymptotics for singular \(n\)-D Burgers' random fields: Gaussian approximation
type: Item
instance of: scholarly article
title: Exact parabolic asymptotics for singular \(n\)-D Burgers' random fields: Gaussian approximation
zbMATH Open document ID: 0928.35214
DOI: 10.1016/S0304-4149(98)00031-3
published in: Stochastic Processes and their Applications
publication date: 1999-11-18T00:00:00Z
review text: An \(n\)-dimensional Burgers' equation with a random initial condition of a gradient type \[ \frac {\partial u}{\partial t} + (u,\nabla)u = \nu \Delta u, \quad u(0,\cdot)=\nabla \xi , \quad t\geq 0, \;x\in \mathbb R^{n}, \] is considered, where \(\nu >0\) and \(u=(u_{1},\ldots ,u_{n})\). The initial velocity potential \(\xi \) is assumed to be a Gaussian zero-mean, measurable, almost surely differentiable, homogeneous and isotropic random field on \(\mathbb R^{n}\) with \(E\xi ^{2}(x) =1\). Let its covariance function be of the form \(L(| x|)/| x| ^{\alpha}\), \(0<\alpha <n\), \(L\) being a locally bounded function slowly varying for large values of \(| x| \). The parabolic scaling limit for \(u\) is known, that is, it is known that the finite-dimensional distributions of the random field \(L^{-1/2}(\sqrt {t})t^{1/2+\alpha /4} u(t,a\sqrt {t})\), \(a\in \mathbb R^{n}\), converge weakly as \(t\to \infty \) to a certain homogeneous Gaussian random field \(X(a)\) whose covariance function may be described explicitly. In the paper under review, the rate of this convergence (in the uniform Kolmogorov distance) is studied.
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 60F99
zbMATH DE Number: 1364491
zbMATH Keywords: Burgers' equation
zbMATH Keywords: Gaussian initial condition
zbMATH Keywords: Kolmogorov distance
zbMATH Keywords: rate of convergence
author: Nikolai N. Leonenko
author: Wojbor A. Woyczyński
MaRDI profile type: Publication
cites work: Stratified structure of the Universe and Burgers' equation -- a probabilistic approach
cites work: Statistical properties of shocks in Burgers turbulence
cites work: Q3735866
cites work: Q4097679
cites work: Q4389468
cites work: Q4840412
cites work: Convergence rate for the approximation of the limit law of weakly interacting particles: Application to the Burgers equation
cites work: Q3357215
cites work: Q4057385
cites work: Q4124504
cites work: Limit Gaussian behavior of the solutions of the multidimensional Burger's equation with weak-dependent initial conditions
cites work: Non-central limit theorems for non-linear functional of Gaussian fields
cites work: Large time behavior of elementary waves of burgers' equation under white noise perturbation
cites work: Gibbs-Cox random fields and Burgers turbulence
cites work: Q4694353
cites work: Q4877203
cites work: An exact rate of convergence in the functional central limit theorem for special martingale difference arrays
cites work: The burgers equation with a noisy force and the stochastic heat equation
cites work: Q4840699
cites work: The partial differential equation ut + uux = μxx
cites work: An extremal rearrangement property of statistical solutions of Burgers' equation
cites work: Limit behavior of quadratic forms of moving averages and statistical solutions of the Burgers' equation
cites work: Q4692964
cites work: Q5534404
cites work: Q4004188
cites work: Sharpness of the normal approximation of functionals of strongly correlated Gaussian random fields
cites work: Q4940333
cites work: Non-Gaussian limit distributions of solutions of the many-dimensional B�rgers equation with random initial data
cites work: Spectral properties of the scaling limit solutions of the Burger’s equation with singular data
cites work: Tauberian and Abelian theorems for correlation function of a homogeneous isotropic random field
cites work: Q4866408
cites work: Scaling limits of solutions of the Burgers equation with singular non-Gaussian data
cites work: On the rate of convergence to the normal law for solutions of the Burgers equation with singular initial data.
cites work: Limiting distributions of the solutions of the many-dimensional B�rgers equation with random initial data. I
cites work: The accuracy of the normal approximation for minimum contrast estimates
cites work: Hyperbolic asymptotics in Burgers' turbulence and extremal processes
cites work: The large-scale structure of the universe and quasi-Voronoi tessellation of shock fronts in forced Burgers turbulence in \(\mathbb{R}^d\)
cites work: Q4327561
cites work: Q3999495
cites work: Remarks on the Burgers Equation
cites work: Scale renormalization and random solutions of the Burgers equation
cites work: Evolution of Burgers' turbulence in the presence of external forces
cites work: Two results concerning asymptotic behavior of solutions of the Burgers equation with force
cites work: Statistics of shocks in solutions of inviscid Burgers equation
cites work: Q4884157
cites work: Burgers' equation by non-local shot noise data
cites work: Q4151274
cites work: Q4296815
cites work: Burgers-KPZ turbulence. Göttingen lectures
Wikidata QID: Q126550341
Recommended article: The Gaussian structure of the singular stochastic Burgers equation
Recommended article: Q4839448
Recommended article: Q4884157
Recommended article: Asymptotics of the density of parabolic Anderson random fields
Recommended article: Non-Gaussian limit distributions of solutions of the many-dimensional B�rgers equation with random initial data
Recommended article: Non-Gaussian limit distributions for solutions of Burgers equation with strongly dependent random initial conditions
Recommended article: Q4866408
Recommended article: Q5284174
description: scientific article; zbMATH DE number 1364491