GEOMETRIC-PROPERTIES OF SOME FAMILIAR DIFFUSIONS IN RN

Consider a convex domain B in 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(n)p(s2, x, y), (s, x, y) is-an-element-of R+ x B2, is concave.