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


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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

Personalised recommendations