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

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