Stochastic Differential and Difference Equations pp 345-353 | Cite as

# On the Interacting Measure-Valued Branching Processes

Conference paper

## 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 }*)**μ*∈*M*(*E*) which uniquely satisfies the following martingale problem: ∀F ∈*bC*^{2}(*R*), ∀*f*∈*D*(*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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## References

- [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 - [D]D. Dawson,
*Measure-Valued Markov Processes*,Lecture Notes of Mathematics,**1541**, Springer-Verlag, Berlin, 1993.Google Scholar - [EM]A. Etheridge and P. March, A note on superprocesses,
*Probab. Th. Rel. Fields*,**89**(1991), 141–147.MathSciNetMATHCrossRefGoogle Scholar - [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 - [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 - [L]Z. H. Li, On the absolute continuity of branching Brownian motion with mean field interaction, preprint, 1996.Google Scholar
- [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 - [P1]E. Perkins, Measure-valued branching diffusions with spatial interactions,
*Probab. Th. Rel. Fields*,**94**(1992), 189–245.MathSciNetMATHCrossRefGoogle Scholar - [P2]E. Perkins,
*Conditional Dawson-Watanabe processes and FlemingViot processes, Seminar on Stochastic Processes*, Birkhäuser, Boston, 1993.Google Scholar - [S]S. Sugitani, Some properties for the measure-valued Branching diffusion processes,
*J. Math. Soc. Japan*,**41**.3 (1989), 437–461.MathSciNetCrossRefGoogle Scholar - [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 - [Z1]X. L. Zhao, Some absolute continuity of superdiffusions and super-stable processes,
*Stoch. Proc. Appl*,**50**(1994), 21–36.MATHCrossRefGoogle Scholar - [Z2]X. L. Zhao, The Absolute Continuity for Interacting Measure-Valued Branching Brownian Motion,
*Chin. Ann. Math.*,**18B**.1 (1997), 4754.Google Scholar - [ZY]X. L. Zhao and M. Yang, A Limit Theorem for Interacting Measure-Valued Branching Processes,
*Acta Mathematica Scientia*,**17B**.1 (1997).MathSciNetGoogle Scholar

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© Springer Science+Business Media New York 1997