Maximal displacement of spectrally negative branching Lévy processes
Christophe Profeta
Abstract
We consider a branching Markov process in continuous time in which the particles evolve independently as spectrally negative Lévy processes. When the branching mechanism is critical or subcritical, the process will eventually die and we may define its overall maximum, i.e. the maximum location ever reached by a particle. The purpose of this paper is to give asymptotic estimates for the survival function of this maximum. In particular, we show that in the critical case the asymptotics is polynomial when the underlying Lévy process oscillates or drifts towards +∞, and is exponential when it drifts towards −∞.
Topics & Concepts
MathematicsBranching processExponential functionLévy processBranching (polymer chemistry)Statistical physicsMarkov processFunction (biology)Mathematical analysisApplied mathematicsCombinatoricsStatisticsPhysicsComposite materialMaterials scienceEvolutionary biologyBiologyStochastic processes and statistical mechanicsProbability and Risk ModelsStochastic processes and financial applications