Catalytic and Mutually Catalytic Branching
نویسندگان
چکیده
In this paper we present an overview of recent work on lattice and measure-valued models of catalytic reactions and in particular catalytic branching systems. The main phenomena exhibited by nearly critical branching systems are dimension-dependent clumping in small and large space and time scales. Special emphasis is given to the e ects which occur when the catalyst is highly clumped and in particular when in the continuum models the catalyst is a time-dependent singular measure. Finally, the interactive model of mutually catalytic branching is described and some recent results are reviewed. The basic tools include log-Laplace functionals, measure-valued martingale problems, collision local times, and duality.
منابع مشابه
Infinite Rate Mutually Catalytic Branching
Consider the mutually catalytic branching process with finite branching rate γ. We show that as γ → ∞, this process converges in finite-dimensional distributions (in time) to a certain discontinuous process. We give descriptions of this process in terms of its semigroup in terms of the infinitesimal generator and as the solution of a martingale problem. We also give a strong construction in ter...
متن کاملMutually Catalytic Amphiphiles: Simulated Chemical Evolution and Implications to Exobiology*
A description of the emergence of life should delineate a chemically rigorous gradual transition from random collections of simple organic molecules to spatially confined assemblies displaying rudimentary self-reproduction capacity. It has been suggested that large sets of mutually catalytic molecules, and not self-replicating information-carrying biopolymers, could have been the precursors of ...
متن کاملIndividuals at the origin in the critical catalytic branching random walk
At the birth moment the newborn particles are located at the origin but after this moment behave independently and stochastically the same as the parent individual. The model we consider here is a particular case of the so-called branching random walk in catalytic medium. The longtime behavior of branching random walks in catalytic media of various types were investigated by a number of authors...
متن کاملThe Longtime Behavior of Branching Random Walk in a Catalytic Medium
We consider catalytic branching random walk (the reactant) where the state space is a countable Abelean group. The branching is critical binary and the local branching rate is given by a catalytic medium. Here the medium is itself an autonomous (ordinary) branching random walk (the catalyst) – maybe with a different motion law. For persistent catalyst (transient motion) the reactant shows the u...
متن کاملInteracting Fisher{wright Diiusions in a Catalytic Medium
We study the longtime behaviour of interacting systems in a randomly uctuating (space{ time) medium and focus on models from population genetics. There are two prototypes of spatial models in population genetics: spatial branching processes and interacting Fisher{Wright diiusions. Quite a bit is known on spatial branching processes where the local branching rate is proportional to a random envi...
متن کامل