Neumann Heat kernel monotonicity

TL;DR

This paper investigates the monotonicity of the diagonal of transition probabilities for reflected Bessel processes in different dimensions, establishing increasing behavior for dimensions greater than two and counterexamples for dimension two.

Contribution

It provides a rigorous proof of monotonicity for d>2 and demonstrates the failure of this property at d=2 for reflected Bessel processes.

Findings

Monotonicity holds for d>2

Monotonicity fails for d=2

Transition probabilities behavior varies with dimension

Abstract

We prove that the diagonal of the transition probabilities for the d-dimensional Bessel processes on (0, 1], reflected at 1, which we denote by $p_R^N(t, r,r)$, is an increasing function of r for d>2 and that this is false for d=2.