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Geometric properties of some familiar diffusions in \({\mathbb{R}{}}^ n\). (English) Zbl 0776.35024
The author considers the equation \(\partial\psi/\partial t={1\over 2}\Delta\psi-V\psi\) on a convex domain in \(n\)-dimensional Euclidean space under zero Dirichlet boundary condition for every \(t\geq 0\). The main result states that for the fundamental solution \(p(t,x,y)\) of the above equation the function \((s,x,y)\mapsto s\ln s^ np(s^ 2,x,y)\) is concave if either \(V\equiv 0\) or \(V\) strictly positive with \(V^{-1/2}\) concave. The proof is based on an inequality of Brunn-Minkowski type and the Feynman-Kac formula. A corollary yields that for \(n\geq 3\) the function \((t,x,y)\mapsto\left(\int^ t_ 0p(s,x,y)ds\right)^{-1/(n- 2)}\) is convex.

MSC:
35K20 Initial-boundary value problems for second-order parabolic equations
60J65 Brownian motion
26A51 Convexity of real functions in one variable, generalizations
60J60 Diffusion processes
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