Monotonicity properties of the Neumann heat kernel in the ball ✩

A well-known conjecture of R. Laugesen and C. Morpurgo asserts that the diagonal of the Neumann heat kernel of the unit ball U ⊂ Rn is a strictly increasing radial function. In this paper we use probabilistic arguments to settle this conjecture and to prove some inequalities for the Neumann heat kernel in the ball. © 2010 Elsevier Inc. All rights reserved.