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- W2009571444 abstract "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 p t ( x , y ) and p̃ t ( x , y ) denote the corresponding parabolic heat kernels for the respective Neumann problems, then: p t (x, y) ≥ p ̃ t (x, y) ∀x, y ∈ D, ∀t > 0. The quantities p t ( x , y ) and p̃ t ( x , y ) can be interpreted as the transition densities of the reflecting Brownian motions X t and X̃ t 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." @default.
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- W2009571444 date "1994-07-01" @default.
- W2009571444 modified "2023-10-17" @default.
- W2009571444 title "Reflecting Brownian Motions and Comparison Theorems for Neumann Heat Kernels" @default.
- W2009571444 doi "https://doi.org/10.1006/jfan.1994.1085" @default.
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