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)


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), \( {F_f}\left( \mu \right)\;F\left( {\left\langle {\left. {\mu ,f} \right\rangle } \right.} \right), \)
$$ {F_{f}}\left( {{X_{t}}} \right)\; - {F_{f}}\left( {{X_{t}}} \right) - \int_{0}^{t} {\left. {\left\langle {{X_{s}},\left( {A + b\left( {\cdot {X_{s}}} \right)} \right)f} \right.} \right\rangle F'\left( {\left. {\left\langle {{X_{s}},f} \right.} \right\rangle } \right) + \left. {{X_{s}}} \right\rangle ,\left\langle {\frac{{c\left( {\cdot ,{X_{s}}} \right)}}{2}{f^{2}}} \right.F''\left( {\left. {\left\langle {{X_{s}},f} \right.} \right\rangle } \right)ds} $$




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  1. [BH]
    H. Begher, and G. C. Hsiao, Nonlinear Boundary Value roblem for a Class of Elliptic Systems, Lecture Notes of Mathematics, Springer-Verlag, Berlin, 1980.Google Scholar
  2. [D]
    D. Dawson, Measure-Valued Markov Processes,Lecture Notes of Mathematics, 1541, Springer-Verlag, Berlin, 1993.Google Scholar
  3. [EM]
    A. Etheridge and P. March, A note on superprocesses, Probab. Th. Rel. Fields, 89 (1991), 141–147.MathSciNetMATHCrossRefGoogle Scholar
  4. [KS]
    N. Konno and T. Shiga, Stochastic partial differential equations for some measure-valued diffusions, Probab, Th. Rel. Fields, 79 (1988), 201–225.MathSciNetMATHCrossRefGoogle Scholar
  5. [LN]
    T. Y. Lee and W. M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc., 333 (1992), 365–378.MathSciNetMATHCrossRefGoogle Scholar
  6. [L]
    Z. H. Li, On the absolute continuity of branching Brownian motion with mean field interaction, preprint, 1996.Google Scholar
  7. [MR]
    M. Méléard and S. Roelly, Interacting branching measure processes, in: Stochastic Partial Differential Equations and Applications (ed. G. Da Prato and L. Tubaro), PRNM 268, Longman Scientific and Technical, Harlow, 1992.Google Scholar
  8. [P1]
    E. Perkins, Measure-valued branching diffusions with spatial interactions, Probab. Th. Rel. Fields, 94 (1992), 189–245.MathSciNetMATHCrossRefGoogle Scholar
  9. [P2]
    E. Perkins, Conditional Dawson-Watanabe processes and FlemingViot processes, Seminar on Stochastic Processes, Birkhäuser, Boston, 1993.Google Scholar
  10. [S]
    S. Sugitani, Some properties for the measure-valued Branching diffusion processes, J. Math. Soc. Japan, 41.3 (1989), 437–461.MathSciNetCrossRefGoogle Scholar
  11. [W]
    B. J. Walsh, An Introduction to Stochastic Partial Differential Equations, Lecture Notes of Mathematics, 1180, Springer—Verlag, New York, 1986, 266–348.Google Scholar
  12. [Z1]
    X. L. Zhao, Some absolute continuity of superdiffusions and super-stable processes, Stoch. Proc. Appl, 50 (1994), 21–36.MATHCrossRefGoogle Scholar
  13. [Z2]
    X. L. Zhao, The Absolute Continuity for Interacting Measure-Valued Branching Brownian Motion, Chin. Ann. Math., 18B.1 (1997), 4754.Google Scholar
  14. [ZY]
    X. L. Zhao and M. Yang, A Limit Theorem for Interacting Measure-Valued Branching Processes, Acta Mathematica Scientia, 17B.1 (1997).MathSciNetGoogle Scholar

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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