Renewing Fleming-Viot-type particle system

We consider a Fleming-Viot-type particle system consisting of independently moving particles that are killed on the boundary of a domain. At the time of death of a particle, another particle branches. We are considering a special case when the particles are driven by Brownian motion. We will show results on the long term behavior of the process.  Using some new tail estimates for the stochastic fixed point equation X = AX + B. we will obtain the law of the iterated logarithm for the process. Joint work with Krzysztof Burdzy.