An example on highly singular parabolic measure
type: Item
instance of: scholarly article
title: An example on highly singular parabolic measure
zbMATH Open document ID: 0726.35050
published in: The Annals of Probability
publication date: 1988-01-01T00:00:00Z
review text: We consider solutions of parabolic equations. \(Lu=a(x,t)u_{xx}-u_ t=0\) in the region \(\{t>0\}\), where \[ (0.1)\;0<\Lambda_ 1\leq a(x,t)\leq \Lambda_ 2<+\infty \] and a is of class \(C^ 2\) in the region; so all solutions are classical. Because \(a(x,t)\leq \Lambda_ 2\) the Dirichlet problem (with bounded, continuous data on the line \(t=0)\) has solutions and these satisfy the maximum principle. The value of a solution at \((x,t)\) can be treated as a linear functional of the boundary values; the value is therefore represented by a probability measure \(\omega^{(x,t)}\) on the boundary - the parabolic measure. Theorem. For each \(\delta>0\) there is some coefficient \(a(x,t)\), \(C^ 2\) in \(\{t>0\}\), satisfying (0.1), for which all parabolic measures are concentrated on a single boundary set of Hausdorff dimension \(<\delta\).
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 31B25
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 4198591
zbMATH Keywords: parabolic measure
zbMATH Keywords: boundary set
author: Robert Kaufman
author: Jang-Mei Wu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991599
OpenAlex ID: W2005228838
DOI: 10.1214/AOP/1176991599
description: scientific article; zbMATH DE number 4198591