# zbMATH — the first resource for mathematics

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
Full Text: