86 research outputs found
Ergodic behavior of locally regulated branching populations
For a class of processes modeling the evolution of a spatially structured
population with migration and a logistic local regulation of the reproduction
dynamics, we show convergence to an upper invariant measure from a suitable
class of initial distributions. It follows from recent work of Alison Etheridge
that this upper invariant measure is nontrivial for sufficiently large
super-criticality in the reproduction. For sufficiently small
super-criticality, we prove local extinction by comparison with a mean field
model. This latter result extends also to more general local reproduction
regulations.Comment: Published at http://dx.doi.org/10.1214/105051606000000745 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
The process of most recent common ancestors in an evolving coalescent
Consider a haploid population which has evolved through an exchangeable
reproduction dynamics, and in which all individuals alive at time have a
most recent common ancestor (MRCA) who lived at time , say. As time goes
on, not only the population but also its genealogy evolves: some families will
get lost from the population and eventually a new MRCA will be established. For
a time-stationary situation and in the limit of infinite population size
with time measured in generations, i.e. in the scaling of population
genetics which leads to Fisher-Wright diffusions and Kingman's coalescent, we
study the process whose jumps form the point process of
time pairs when new MRCAs are established and when they lived. By
representing these pairs as the entrance and exit time of particles whose
trajectories are embedded in the look-down graph of Donnelly and Kurtz (1999)
we can show by exchangeability arguments that the times as well as the
times from a Poisson process. Furthermore, the particle representation
helps to compute various features of the MRCA process, such as the distribution
of the coalescent at the instant when a new MRCA is established, and the
distribution of the number of MRCAs to come that live in today's past
Hierarchical equilibria of branching populations
The objective of this paper is the study of the equilibrium behavior of a population on the hierarchical group (Omega)N consisting of families of individuals undergoing critical branching random walk and in addition these families also develop according to a critical branching process. Strong transience of the random walk guarantees existence of an equilibrium for this two-level branching system. In the limit N -> (infinity symbol) (called the hierarchical mean field limit), the equilibrium aggregated populations in a nested sequence of balls (symbole)(N) of hierarchical radius (symbol) converge to a backward Markov chain on R+. This limiting Markov chain can be explicitly represented in terms of a cascade of subordinators which in turn makes possible a description of the genealogy of the population.Multilevel branching, hierarchical mean-field limit, strong transience,genealogy.
Hierarchical equilibria of branching populations
The objective of this paper is the study of the equilibrium behavior of a
population on the hierarchical group consisting of families of
individuals undergoing critical branching random walk and in addition these
families also develop according to a critical branching process. Strong
transience of the random walk guarantees existence of an equilibrium for this
two-level branching system. In the limit (called the hierarchical
mean field limit), the equilibrium aggregated populations in a nested sequence
of balls of hierarchical radius converge to a backward
Markov chain on . This limiting Markov chain can be explicitly
represented in terms of a cascade of subordinators which in turn makes possible
a description of the genealogy of the population.Comment: 62 page
Branching systems with long living particles at the critical dimension
A spatial branching process is considered in which particles have a life time law with a tail index smaller than one. Other than in classical branching particle systems, at the critical dimension the system does not suffer local extinction when started from a spatially homogenous initial population. In fact, persistent convergence to a mixed Poissonian system is shown. The random limiting intensity is characterized in law by the random density in a space point of a related age-dependent superprocess at a fixed time. The proof relies on a refined study of the system starting from asymptotically large but finite initial populations
The spatial Lambda-Fleming–Viot process: An event-based construction and a lookdown representation
Positive contraction mappings for classical and quantum Schrodinger systems
The classical Schrodinger bridge seeks the most likely probability law for a
diffusion process, in path space, that matches marginals at two end points in
time; the likelihood is quantified by the relative entropy between the sought
law and a prior, and the law dictates a controlled path that abides by the
specified marginals. Schrodinger proved that the optimal steering of the
density between the two end points is effected by a multiplicative functional
transformation of the prior; this transformation represents an automorphism on
the space of probability measures and has since been studied by Fortet,
Beurling and others. A similar question can be raised for processes evolving in
a discrete time and space as well as for processes defined over non-commutative
probability spaces. The present paper builds on earlier work by Pavon and
Ticozzi and begins with the problem of steering a Markov chain between given
marginals. Our approach is based on the Hilbert metric and leads to an
alternative proof which, however, is constructive. More specifically, we show
that the solution to the Schrodinger bridge is provided by the fixed point of a
contractive map. We approach in a similar manner the steering of a quantum
system across a quantum channel. We are able to establish existence of quantum
transitions that are multiplicative functional transformations of a given Kraus
map, but only for the case of uniform marginals. As in the Markov chain case,
and for uniform density matrices, the solution of the quantum bridge can be
constructed from the fixed point of a certain contractive map. For arbitrary
marginal densities, extensive numerical simulations indicate that iteration of
a similar map leads to fixed points from which we can construct a quantum
bridge. For this general case, however, a proof of convergence remains elusive.Comment: 27 page
Discrete-time classical and quantum Markovian evolutions: Maximum entropy problems on path space
The theory of Schroedinger bridges for diffusion processes is extended to
classical and quantum discrete-time Markovian evolutions. The solution of the
path space maximum entropy problems is obtained from the a priori model in both
cases via a suitable multiplicative functional transformation. In the quantum
case, nonequilibrium time reversal of quantum channels is discussed and
space-time harmonic processes are introduced.Comment: 34 page
- …