Coupled Brownian motions and partial domain monotonicity for the Neumann heat kernel

Abstract

AbstractLet D and Ω be two convex domains (with smooth boundary) and with corresponding Neumann heat kernels ηD(x, y, t) and ηgW(x, y, t). Chavel has conjectured that if x, y ϵ D ⊂ Ω then the following monotonicity result holds for all t: ηD(x, y, t) ⩾ ηΩ(x, y, t). He has proved this in the special case when Ω is a ball centred at either x or y. By exploiting the connection of the Neumann kernel to reflecting Brownian motion it is possible to prove the result when Ω is general and D is a ball centred at either x or y. The proof depends on a careful coupling construction of the reflecting Brownian motions for D and Ω using the same probability space for both processes