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A note on parabolic convexity and heat conduction. (English) Zbl 0854.60058
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 \(C (\overline R_+; R^n)\).

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J65 Brownian motion
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