A domain monotonicity property of the Neuman heat kernel (Q1333593)

The paper deals with the conjecture that a solution to the Neuman problem for the heat equation is monotone with respect to the domain. That is, if \(\Omega \subset D\), then \(p^ \Omega \geq p^ D\), where \(p\) denotes the solution. The conjecture was proved to be false for general domains by Bass and Burdzy. In this paper, by a probabilistic approach employing reflected Brownian motion, it is shown that the conjecture holds for some particular domains.