### 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.

Original language | English (US) |
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Pages (from-to) | 109-128 |

Number of pages | 20 |

Journal | Journal of Functional Analysis |

Volume | 123 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1994 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Analysis

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## Cite this

*Journal of Functional Analysis*,

*123*(1), 109-128. https://doi.org/10.1006/jfan.1994.1085