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If ξ(t) is the solution of homogeneous SDE in R m, and T ∃ is the first exit moment of the process from a small domain D ∃, then the total expansion for the following functional showing independence of the exit time and exit place is $$Eexp(-\lambda T_{\epsilon })f\left(\frac{\xi \left(T_{\epsilon }\right)}{\epsilon }\right)-Eexp(-\lambda T_{\epsilon })Ef\left(\frac{\xi \left(T_{\epsilon }\right)}{\epsilon }\right),\epsilon \searrow 0,\lambda >0.$$

We have random number of independent diffusion processes with absorption on boundaries in some region at initial time t = 0. The initial numbers and positions of processes in region is defined by the Poisson random measure. It is required to estimate the number of the unabsorbed processes for the fixed time τ > 0. The Poisson random measure depends on τ and τ → ∞.