Limit theorems for number of diffusion processes, which did not absorb by boundaries (Q861588)

A random number of diffusion processes are initiated in a bounded region with initial conditions distributed according to a Poisson random measure depending on a parameter \(\tau > 0\). These are killed on hitting the boundary. Under the condition that the boundary is a Lyapunov surface, it is shown that the processes not killed by time \(\tau\) have a Poisson distribution with a parameter \(a(\tau)\) for which an expression is derived. Under the additional assumption that a certain scaling limit exists as \(\tau \rightarrow \infty\), the limiting Poisson distribution as \(\tau \rightarrow \infty\) is shown to exist and is characterized.