2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/166001A well known conjecture of R. Laugesen and C. Morpurgo asserts that the diagonal element of the Neumann heat kernel of the unit ball in $\mathbb{R}^{n}$ ($n\geq1$) is a radially increasing function. In this paper, we use probabilistic arguments to settle this conjecture, and, as an application, we derive a new proof of the Hot Spots conjecture of J. Rauch in the case of the unit disk.14 pages, 1 figureProbabilityAnalysis of PDEs60J65, 60J35text