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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 $$C (\overline R_+; R^n)$$.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60J65 Brownian motion
##### Keywords:
parabolic convexity; Brownian motion; Ehrhard inequality
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