A Ray-Knight theorem for symmetric Markov processes

Lecture 1: Generalized Ray Knight Theorem for Finite Markov Chains

Binary trees, exploration processes, and an extented Ray–Knight Theorem

We study the bijection betwen binary Galton Watson trees in continuous time and their exploration process, both in the suband in the supercritical cases. We then take the limit over renormalized quantities, as the size of the population tends to infinity. We thus deduce Delmas’ generalization of the second Ray–Knight theorem.

Branching processes with competition and generalized Ray Knight Theorem

We consider a discrete model of population with interaction where the birth and death rates are non linear functions of the population size. After proceeding to renormalization of the model parameters, we obtain in the limit of large population that the population size evolves as a diffusion solution of the SDE

Binary Trees, Exploration Processes, and an Extended Ray-Knight Theorem

We study the bijection between binary Galton–Watson trees in continuous time and their exploration process, both in the subcritical and in the supercritical cases. We then take the limit over renormalized quantities, as the size of the population tends to ∞. We thus deduce Delmas’ generalization of the second Ray–Knight theorem.

A direct proof of the Ray-Knight theorem

© Springer-Verlag, Berlin Heidelberg New York, 1981, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impressio...