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On the coupling property of Lévy processes

René L. Schilling, Jian Wang (2011)

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

We give necessary and sufficient conditions guaranteeing that the coupling for Lévy processes (with non-degenerate jump part) is successful. Our method relies on explicit formulae for the transition semigroup of a compound Poisson process and earlier results by Mineka and Lindvall–Rogers on couplings of random walks. In particular, we obtain that a Lévy process admits a successful coupling, if it is a strong Feller process or if the Lévy (jump) measure has an absolutely continuous component.

On the equivalence of some eternal additive coalescents

Anne-Laure Basdevant (2008)

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

In this paper, we study additive coalescents. Using their representation as fragmentation processes, we prove that the law of a large class of eternal additive coalescents is absolutely continuous with respect to the law of the standard additive coalescent on any bounded time interval.

On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics

Nalini Anantharaman (2004)

Journal of the European Mathematical Society

We study the zero-temperature limit for Gibbs measures associated to Frenkel–Kontorova models on ( ℝ d ) ℤ / ℤ d . We prove that equilibrium states concentrate on configurations of minimal energy, and, in addition, must satisfy a variational principle involving metric entropy and Lyapunov exponents, a bit like in the Ruelle–Pesin inequality. Then we transpose the result to certain continuous-time stationary stochastic processes associated to the viscous Hamilton–Jacobi equation. As the viscosity vanishes, the...

Optimal stopping of a risk process

Elżbieta Ferenstein, Andrzej Sierociński (1997)

Applicationes Mathematicae

Optimal stopping time problems for a risk process U t = u + c t - ∑ n = 0 N ( t ) X n where the number N(t) of losses up to time t is a general renewal process and the sequence of X i ’s represents successive losses are studied. N(t) and X i ’s are independent. Our goal is to maximize the expected return before the ruin time. The main results are closely related to those obtained by Boshuizen and Gouweleew [2].

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