Limit theorems for Markov processes indexed by continuous time Galton--Watson trees

We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton--Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time t. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. The latter has the same generator as the Markov process along the branches plus additional jumps, associated with branching events of accelerated rate and biased distribution. This comes from the fact that choosing an individual uniformly at time t favors lineages with more branching events and larger offspring number. The central limit theorem is considered on a special case. Several examples are developed, including applications to splitting diffusions, cellular aging, branching L\'{e}vy processes.Comment: Published in at http://dx.doi.org/10.1214/10-AAP757 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Weibull-type limiting distribution for replicative systems

The Weibull function is widely used to describe skew distributions observed in nature. However, the origin of this ubiquity is not always obvious to explain. In the present paper, we consider the well-known Galton-Watson branching process describing simple replicative systems. The shape of the resulting distribution, about which little has been known, is found essentially indistinguishable from the Weibull form in a wide range of the branching parameter; this can be seen from the exact series expansion for the cumulative distribution, which takes a universal form. We also find that the branching process can be mapped into a process of aggregation of clusters. In the branching and aggregation process, the number of events considered for branching and aggregation grows cumulatively in time, whereas, for the binomial distribution, an independent event occurs at each time with a given success probability.Comment: 6 pages and 5 figure

BES Results on Charmonium Decays and Transitions

Results are reported based on samples of 58 million $J/\psi$ and 14 million $\psi(2S)$ decays obtained by the BESII experiment. Improved branching fraction measurements are determined, including branching fractions for $J/\psi\to\pi^+\pi^-\pi^0$, $\psi(2S)\to \pi^0 J/\psi$, $\eta J/\psi$, $\pi^0 \pi^0 J/\psi$, anything $J/\psi$, and \psi(2S)\to\gamma\chi_{c1},\gamma\chi_{c2}\to\gamma\gamma\jpsi. Using 14 million $\psi(2S)$ events, $f_0(980)f_0(980)$ production in $\chi_{c0}$ decays and $K^*(892)^0\bar K^*(892)^0$ production in $\chi_{cJ}~(J=0,1,2)$ decays are observed for the first time, and branching ratios are determined.Comment: Parallel Talk presented at ICHEP04. 4 pages and 6 figure

Asymmetric Time Evolution And Indistinguishable Events

With a time asymmetric theory, in which quantum mechanical time evolution is given by a semigroup of operators rather than by a group, the states of open systems are represented by density operators exhibiting a branching behavior. To treat the indistinguishably of the members of experimental ensembles, we hypothesize that environmental interference occurs during events that are themselves fundamentally indistinguishable.Center for Complex Quantum System

Dynamical scaling in branching models for seismicity

We propose a branching process based on a dynamical scaling hypothesis relating time and mass. In the context of earthquake occurrence, we show that experimental power laws in size and time distribution naturally originate solely from this scaling hypothesis. We present a numerical protocol able to generate a synthetic catalog with an arbitrary large number of events. The numerical data reproduce the hierarchical organization in time and magnitude of experimental inter-event time distribution.Comment: 3 figures to appear on Physical Review Letter