317 research outputs found
Transience and recurrence of random walks on percolation clusters in an ultrametric space
We study existence of percolation in the hierarchical group of order ,
which is an ultrametric space, and transience and recurrence of random walks on
the percolation clusters. The connection probability on the hierarchical group
for two points separated by distance is of the form , with , non-negative constants , and . Percolation was proved in Dawson and Gorostiza
(2013) for , with
. In this paper we improve the result for the critical case by
showing percolation for . We use a renormalization method of the type
in the previous paper in a new way which is more intrinsic to the model. The
proof involves ultrametric random graphs (described in the Introduction). The
results for simple (nearest neighbour) random walks on the percolation clusters
are: in the case the walk is transient, and in the critical case
, there exists a critical
such that the walk is recurrent for and transient for
. The proofs involve graph diameters, path lengths, and
electric circuit theory. Some comparisons are made with behaviours of random
walks on long-range percolation clusters in the one-dimensional Euclidean
lattice.Comment: 27 page
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
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.
Functional Limit Theorems for Occupation Time Fluctuations of Branching Systems in the Cases of Large and Critical Dimensions
Functional central limit theorem; Occupation time fluctuation; Branching particle system; Generalized Wiener process; Critical dimension
A Long Range Dependence Stable Process and an Infinite Variance Branching System
We prove a functional limit theorem for the rescaled occupation time fluctuations of a (d, , )- branching particle system (particles moving in Rd according to a symmetric -stable L´evy process, branching law in the domain of attraction of a (1 + )-stable law, 0 d/(d + ), which coincides with the case of finite variance branching ( = 1), and another one for d/(d + ), where the long range dependence depends on the value of . The long range dependence is characterized by a dependence exponent which describes the asymptotic behavior of the codierence of increments of on intervals far apart, and which is d/ for the first case and (1 + - d/(d + ))d/ for the second one. The convergence proofs use techniques of S0(Rd)-valued processes.Branching particle system, occupation time fluctuation, functional limit theorem, stable process, long range dependence.
Occupation Time Fluctuations of an Infinite Variance Branching System in Large Dimensions
We prove limit theorems for rescaled occupation time fluctuations of a (d, , )-branching particle system (particles moving in Rd according to a spherically symmetric -stable L´evy process, (1 + )- branching, 0 (1 + )/. The fluctuation processes are continuous but their limits are stable processes with independent increments, which have jumps. The convergence is in the sense of finite-dimensional distributions, and also of space-time random fields (tightness does not hold in the usual Skorohod topology). The results are in sharp contrast with those for intermediate dimensions, /Branching particle system, critical and large dimensions, limit theorem, occupation time fluctuation, stable process.
Functional Limit Theorems for Occupation Time Fluctuations of Branching Systems in the Case of Long-Range Dependence
Functional central limit theorem; Occupation time
uctuation; Branching particle system; Distribution-valued Gaussian process; Fractional Brownian motion; Sub-fractional Brownian motion; Long-range dependence
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