A criterion of convergence of measure‐valued processes: application to measure branching processes

  title={A criterion of convergence of measure‐valued processes: application to measure branching processes},
  author={Sylvie Roelly‐ Coppoletta},
  journal={Stochastics An International Journal of Probability and Stochastic Processes},
  • Sylvie Roelly‐ Coppoletta
  • Published 1 April 1986
  • Mathematics
  • Stochastics An International Journal of Probability and Stochastic Processes
In this paper martingale properties of a Measure Branching process are investigated. Uniqueness and continuity of this process are proven by a martingale approach. For the existence, we approximate the measure branching process by a sequence of infinite particle branching diffusion processes, and show the convergence in distribution by a new criterion for measure‐valued processes. We also give properties about local structure of the process. 

Interacting measure branching processes. Some bounds for the support

We prove a tightness result for a sequence of interacting branching-diffusion processes. The limit points are continuous measure-valued processes and satisfy a martingale property. We state that they

Discontinuous Measure-Valued Branching Processes and Generalized Stochastic Equations

We study a class of integrable and discontinuous measure-valued branching processes. They are constructed as limits of renormalized spatial branching processes, the underlying branching distribution


This is a survey on the theory of measure-valued branching pro- cesses (Dawson-Watanabe superprocesses) and their associated immigration processes formulated by skew convolution semigroups. The

Comparisons for measure valued processes with interactions

This paper considers some measure-valued processes {Xt\dvtxt∈[0,T]} based on an underlying critical branching particle structure with random branching rates. In the case of constant branching these

On Extinction of Measure-Valued Markov Processes

We begin with introducing superprocesses with branching rate functional and historical superprocesses. We consider the notions of recurrence, transience and extinction property of measure-valued

A Convergence Criterion for Measure-Valued Processes, and Application to Continuous Superprocesses

A criterion for weak convergence of measure-valued processes is proved, and it is exemplified by showing convergence of branching particle systems to continuous superprocesses.

Absolute Continuity of the Measure States in a Branching Model with Catalysts

Spatially homogeneous measure-valued branching Markov processes X on the real line ℝ with certain motion processes and branching mechanisms with finite variances have absolutely continuous states

Limit Theorems for Continuous-Time Branching Flows

Some scaling limit theorems for the flow are proved, which lead to the path-valued branching processes and nonlocal branching superprocesses over the positive half line studied in Li (2012).



The critical measure diffusion process

SummaryA multiplicative stochastic measure diffusion process is the continuous analogue of an infinite particle branching diffusion process. In this paper the limiting behavior of the critical

Stopping Times and Tightness. II

To establish weak convergence of a sequence of martingales to a continuous martingale limit, it is sufficient (under the natural uniform integrability condition) to establish convergence of

A weighted occupation time for a class of measured-valued branching processes

SummaryA weighted occupation time is defined for measure-valued processes and a representation for it is obtained for a class of measure-valued branching random motions on Rd. Considered as a process

Bessel diffusions as a one-parameter family of diffusion processes

O. Introduction By a Bessel diffusion process with index ~ (~ > 0), we mean a conservative onedimensional diffusion process on [0, o0) determined by the local generator

Convergence of Probability Measures

  • P. M. Lee
  • Mathematics
    The Mathematical Gazette
  • 1970
Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.