Fix some $r >0$ and let $\mathcal P$ be a unit intensity Poisson point process on $\mathbb R^d - \mathbb B(0,r)$. Let $W\_t = \cup\_{s \leq t} \mathbb B(B\_t,r)$ be the Brownian sausage around a Brownian motion $B\_t$ started from $\mathbf 0$. Run the process until the time $\tau = \inf \{ t \colon W\_t \cap \mathcal P \neq \emptyset\}$ that the sausage hits a point in $\mathcal P$. 


Now, let $\mathcal P'$ be an independent unit intensity Poisson point process. Define the set $$\mathcal P'' = (\mathcal P - W\_\tau) \cup (\mathcal P' \cap W\_\tau).$$ So we are taking out the point that $W\_\tau$ hit and putting back in $W\_\tau \cap \mathcal P'$. 


Is $\mathcal P''$ a unit intensity Poison point process on $\mathbb R^d$? 



Brownian sausage surgery of Poisson point process
See the appendix of this paper on the Brownian frog model: <https://arxiv.org/abs/1710.05811>.


