We give a pathwise construction of two-parameter family purely-atomic-measure-valued diffusions in which ranked masses atoms are stationary with the Poisson–Dirichlet(α,θ) distributions, for α∈(0,1) and θ≥0. These processes resolve conjecture Feng Sun (Probab. Theory Related Fields 148 (2010) 501–525). build on our previous work (α,0)- (α,α)-interval partition evolutions. The extension to gener...

Completing their initial phase of rapid growth social media networks are expected to reach a plateau from where on they are in a statistically stationary state. Such stationary states may have different dynamical properties. For example, if each message in a network is followed by a reply in opposite direction, the dynamics is locally balanced. Otherwise, if messages are ignored or forwarded to...

The effects of random forces on the emergence of singularities in the Navier-Stokes equations are investigated. In spite of the presence of white noise, the paths of a martingale suitable weak solution have a set of singular points of one-dimensional Hausdorff measure zero. Furthermore statistically stationary solutions with finite mean dissipation rate are analysed. For these stationary soluti...

The classical result on singularities for the 3D Navier-Stokes equations says that the 1-dimensional Hausdorff measure of the set of singular points is zero. For a stochastic version of the equation, new results are proved. For statistically stationary solutions, at any given time t, with probability one the set of singular points is empty. The same result is true for a.e. initial condition wit...

The graph-related symmetries of a reaction network give rise to certain special equilibria (such as complex balanced equilibria) in deterministic models of dynamics of the reaction network. Correspondingly, in the stochastic setting, when modeled as a continuous-time Markov chain, these symmetries give rise to certain special stationary measures. Previous work by Anderson, Craciun and Kurtz ide...

In some long term studies, a series of dependent and possibly truncated lifetime data may be observed. Suppose that the lifetimes have a common continuous distribution function F. A popular stochastic measure of the distance between the density function f of the lifetimes and its kernel estimate fn is the integrated square error (ISE). In this paper, we derive a central limit theorem for t...