Scaling limits of solutions of the heat equation for singular non-Gaussian data
type: Item
instance of: scholarly article
title: Scaling limits of solutions of the heat equation for singular non-Gaussian data
zbMATH Open document ID: 0926.60054
DOI: 10.1023/A:1023060625577
published in: Journal of Statistical Physics
publication date: 1999-11-21T00:00:00Z
review text: Let \(\{u(t,x), t\geq 0,x\in \mathbb{R}\}\) be the solution of the one-dimensional heat equation with an initial condition of the form \(G(\xi(x))\), where \(\xi(x)\) is a zero-mean stationary Gaussian process with variance 1 and \(G\) is a nonlinear function which satisfies \(E(G^2(\xi(0)))<\infty\). The authors show the weak convergence of the finite-dimensional distributions of the rescaled random field \({1\over B(\sqrt T)} [u(tT,x \sqrt T)-E(G (\xi(0)))]\), as \(T\) tends to infinity, to the finite-dimensional distributions of a random field \(Z_m(t,x)\), assuming that the correlation function \(B(| x|)\) of \(\xi(x)\) satisfies some suitable conditions. The random field \(Z_m(t,x)\) has a spectral representation in terms of multiple stochastic integrals of order \(m\geq 1\), where the integer \(m\), called the Hermitian rank of \(G\), is the first nonzero coefficient in the Chebyshev-Hermite series expansion of \(G-E(G(\xi(0)))\).
reviewed by: David Nualart
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 1279468
zbMATH Keywords: heat equation
zbMATH Keywords: non-Gaussian initial condition
zbMATH Keywords: scaling limit
zbMATH Keywords: multiple stochastic integrals
author: Nikolai N. Leonenko
author: Wojbor A. Woyczyński
MaRDI profile type: Publication
full work available at URL: a:1023060625577
OpenAlex ID: W29525048
description: scientific article; zbMATH DE number 1279468