Geometric Properties of Some Familiar Diffusions in $\mathbb{R}^n$

Abstract

Consider a convex domain $B$ in $\mathbb{R}^n$ and denote by $p(t, x, y)$ the transition probability density of Brownian motion in $B$ killed at the boundary of $B$. The main result in this paper, in particular, shows that the function $s \ln s^np(s^2, x, y), (s, x, y) \in \mathbb{R}_+ \times B^2$, is concave.