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The evolution and Poisson kernels on nilpotent meta-abelian groups

Richard Penney, Roman Urban (2013)

Studia Mathematica

Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to ℝ k , k>1. We consider a class of second order left-invariant differential operators on S of the form ℒ α = L a + Δ α , where α ∈ ℝ k , and for each a ∈ ℝ k , L a is left-invariant second order differential operator on N and Δ α = Δ - ⟨ α , ∇ ⟩ , where Δ is the usual Laplacian on ℝ k . Using some probabilistic techniques (e.g., skew-product formulas for diffusions on S and N respectively) we obtain an...

The exit path of a Markov chain with rare transitions

Olivier Catoni, Raphaël Cerf (2010)

ESAIM: Probability and Statistics

We study the exit path from a general domain after the last visit to a set of a Markov chain with rare transitions. We prove several large deviation principles for the law of the succession of the cycles visited by the process (the cycle path), the succession of the saddle points gone through to jump from cycle to cycle on the cycle path (the saddle path) and the succession of all the points gone through (the exit path). We estimate the time the process spends in each cycle of the cycle path...

The falling apart of the tagged fragment and the asymptotic disintegration of the brownian height fragmentation

Gerónimo Uribe Bravo (2009)

Annales de l'I.H.P. Probabilités et statistiques

We present a further analysis of the fragmentation at heights of the normalized brownian excursion. Specifically we study a representation for the mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable subordinator and use it to study its jumps; this accounts for a description of how a typical fragment falls apart. These results carry over to the height fragmentation of the stable tree. Additionally, the sizes of the fragments in the brownian height fragmentation when it is...

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