Monotonicity properties of the Neumann heat kernel in the ball (Q610770)

The main result of the paper establishes that \(p(t, x, x) < p(t, y, y)\) for \(t > 0\) and \(\| x \|<\|y\| \leq 1\), where \(p\) is the heat kernel for the Laplacian with Neumann boundary conditions on the unit ball in \(\mathbb R^n\) (a conjecture of \textit{R. S. Laugesen} and \textit{C. Morpurgo} [J. Funct. Anal. 155, No.~1, 64--108 (1998; Zbl 0917.47018)]). The proof uses the mirror coupling of reflecting Brownian motions in the unit ball.