Reflecting Brownian Motions and Comparison Theorems for Neumann Heat Kernels

We consider the challenging problem of Chavel′s conjecture on the domain monotonicity of the fundamental solution of the Neumann problem. It says that if D ⊂ D̃ are open convex domains and if pt(x, y) and p̃t(x, y) denote the corresponding parabolic heat kernels for the respective Neumann problems, then: pt(x, y) ≥ p̃t(x, y) ∀x, y ∈ D, ∀t > 0. The quantities pt(x, y) and p̃t(x, y) can be interpreted as the transition densities of the reflecting Brownian motions Xt and X̃t in D and D̃, respectively. The conjecture can be restated as a comparison problem for Brownian motions with reflecting boundary conditions. We give a detailed analysis of the small time asymptotics of the Brownian paths reflected on the boundary of a polyhedron and we give a probabilistic proof of Chavel′s conjecture for small time t uniformly for all x and y in D when the closure of D is contained in D̃. An interesting by-product of our proof is that it does not require the large domain to be convex.