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 \subset \tilde{D}) are open convex domains and if (p_{t}(x, y)) and (\tilde{p}_{t}(x, y)) denote the corresponding parabolic heat kernels for the respective Neumann problems, then:
[
p_{t}(x, y) \geqslant \tilde{p}_{s}(x, y) \quad \forall x, y \in D, \quad \forall t>0
]
The quantities (p_{l}(x, y)) and (\tilde{p}{l}(x, y)) can be interpreted as the transition densities of the reflecting Brownian motions (X{t}) and (\tilde{X}_{t}) in (D) and (\tilde{D}), 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 (\tilde{D}). An interesting by-product of our proof is that it does not require the large domain to be convex. C 1994 Academic Press, Inc.