A note on parabolic convexity and heat conduction
type: Item
instance of: scholarly article
title: A note on parabolic convexity and heat conduction
zbMATH Open document ID: 0854.60058
author: Christer Borell
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1997-01-13T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77540
review text: A subset \(E\) of \(R_+ \times R^n\) is said to be parabolically convex, if for any \(\zeta_0 = (t_0, x_0)\) and \(\zeta_1 = (t_1, x_1)\) belonging to \(E\), \[ \biggl( \bigl( (1 - \theta) \sqrt {t_0} + \theta \sqrt {t_1} \bigr)^2,\;(1 - \theta) x_0 + \theta x_1 \biggr) \in E, \quad 0 \leq \theta \leq 1. \] Let \(D\) be a domain in \(R \times R^n\). The set \(B (\zeta_0, r) = \{\zeta \in D;\;p (\zeta, \zeta_0) < r\}\) is called a heat ball in \(D\) with center at \(\zeta_0\), where \(p\) is a Green function on \(D \times D\) of the heat operator in \(D\) equipped with Dirichlet boundary condition zero. The main result of the paper is Theorem 1.1. If the set \(D \cap \{t > 0\}\) is parabolically convex, then any heat ball in \(D\) with its centre in \(D \cap \{t > 0\}\) is parabolically convex. In the proof of Theorem 1.1 the Brownian motion and Ehrhard inequality are used. Let \(\Phi\) be the standard normal distribution function, \(\beta = (\beta (t))_{t \geq 0}\) denote the normalized Brownian motion in \(R^n\) and suppose \(\mu = P_\beta\) is Wiener measure on \(C (\overline R_+; R^n)\), the space of all continuous mappings of \(\overline R_+\) into \(R^n\) equipped with the topology of uniform convergence of compacts. The Ehrhard inequality of the Brunn-Minkowski type states that \[ \Phi^{-1} \biggl( \mu \bigl( (1 - \theta) B_0 \bigr) \biggr) \geq (1 - \theta) \Phi^{-1} \bigl (\mu (B_0) \bigr) + \theta \Phi^{-1} \bigl( \mu (B_1) \bigr) \] for every \(0 \leq \theta \leq 1\), and every convex Borel sets \(B_0\) and \(B_1\) in \(C (\overline R_+; R^n)\).
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 898067
zbMATH Keywords: parabolic convexity
zbMATH Keywords: Brownian motion
zbMATH Keywords: Ehrhard inequality
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 898067