The shape of the Neumann heat kernel p(t,x,y) on the unit ball is more easily visualized than analyzed. (Dirichlet conditions are better understood.) And so we examine a conjecture restricted to the spatial diagonal: for fixed t, is p(t,x,x) increasing as a function of r=|x|? That is, for reflected Brownian motion in the ball, does the probability of returning to one’s initial point get higher when that initial point is located closer to the boundary?
Laugesen and Morpurgo raised the conjecture 30 years ago and then Pascu and Gageonea proved it 15 years ago for Euclidean space, by an ingenious application of probabilistic mirror coupling (a technique used by Atar and Burdzy for hot spots). We extend the theorem to geodesic balls in hyperbolic space and the hemisphere.
As a corollary, we deduce monotonicity wrt curvature of the Neumann spectral zeta function on a geodesic disk of fixed area — a result known previously only in the large-parameter limit, by Bandle’s theorem on the first nonzero eigenvalue.
[Joint with Jing Wang, Purdue University.]
