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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

Polish Space Absolute Continuity Stochastic Partial Differential Equation Martingale Problem Kronecker Delta Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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