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) ≥ p ̃t(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.
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