On the Interacting Measure-Valued Branching Processes

  • Xuelei Zhao
Conference paper
Part of the Progress in Systems and Control Theory book series (PSCT, volume 23)

Abstract

Let (E, ε) be a Polish space. Denote by M (E) the family of finite measures on E, and equip M (E) with the usual weak convergence topology. Let us first briefly introduce the approximating processes (μ t ) t >0, called interacting branching diffusion processes. For each time t, μ t is a random measure which models branching and diffusing particles in the following way: \( {\mu _t}\; = \sum\nolimits_{i \in {I_t}} {{\delta _{x_t^i}}} \)where I t is the set of indexes of particles alive at time t,δ x is the Kronecker delta function at point x, and x t i are the locations of particles indexed i at time t. Their dynamics are the following: μ 0 is a finite measure on E, describing the initial configuration of the system. Each particle moves following a homogeneous conservative Feller process with generator (A,D (A)), and after a certain lifetime, it vanishes at the location x with the death rate λ(x, μ t) and is replaced by a random number of children. The reproduction law depends on the state of the system by μ t and the location x. This kind of interaction is well known in infinite particle systems. Under suitable hypothesis this interacting diffusion process μ t approximates to the measure-valued branching diffusion processes (X t , P µ ) μM (E) which uniquely satisfies the following martingale problem: ∀F ∈ bC 2(R), ∀fD (A), \( {F_f}\left( \mu \right)\;F\left( {\left\langle {\left. {\mu ,f} \right\rangle } \right.} \right), \)
$$ {F_{f}}\left( {{X_{t}}} \right)\; - {F_{f}}\left( {{X_{t}}} \right) - \int_{0}^{t} {\left. {\left\langle {{X_{s}},\left( {A + b\left( {\cdot {X_{s}}} \right)} \right)f} \right.} \right\rangle F'\left( {\left. {\left\langle {{X_{s}},f} \right.} \right\rangle } \right) + \left. {{X_{s}}} \right\rangle ,\left\langle {\frac{{c\left( {\cdot ,{X_{s}}} \right)}}{2}{f^{2}}} \right.F''\left( {\left. {\left\langle {{X_{s}},f} \right.} \right\rangle } \right)ds} $$
(1.1)

Keywords

Covariance 

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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Xuelei Zhao
    • 1
  1. 1.Institute of MathematicsShantou UniversityShantouChina

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