We study the asymptotic behavior of Brownian motion and its conditioned process in cones using an in®nite series representation of its transition density. A concise probabilistic interpretation of this series in terms of the skew product decomposition of Brownian motion is derived and used to show properties of the transition density.

We derive estimates for the principal eigenvalue of the boundary value problem ∆u = λ(α) u in Ω, ∂u ∂ν = α u on ∂Ω, with α > 0 and Ω ⊂ R n a bounded domain. In the context of superconductivity, our results show the increase of the critical temperature in zero fields for systems with enhanced surface superconductivity. In term of long time behavior of a Brownian motion with creation of particles at the boundary, our study gives estimates for the expected number of particles inside the domain. (2000). 82D55, 35P15, 49G05. Mathematics Subject Classification

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