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 $\mathbb R^d$?

the pointthat $W_\tau$ hit. $\endgroup$