The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail.
Just to avoid misunderstanding, I point out that it is about Poisson point process and not about usual Poisson process.
Addition: here is what I mean exactly. Let's say we have a process $N$ on $\mathbb{R} _+\times \mathbb{R}^d$, $\mathbb{R} _+$ is time, with the intensity measure being the Lebesgue measure on $\mathbb{R} _+\times \mathbb{R}^d$. Let $\mathscr{F} _t$ be the minimal $\sigma$ -algebra containing all random variable $N(Q,U)$, where $Q \in \mathscr{B}([0;T])$, $U \in \mathscr{B}(\mathbb{R}^d)$. Or we may take minimal complete right-continuous $\sigma$ - algebra with this property. Let $\tau$ be a stopping time with respect to $(\mathscr{F} _t)$. It seems reasonable to conjecture, that the process $\bar N $ defined by
$$\bar N ([0;s],U) = N ([\tau;\tau + s],U) - N ([0;\tau],U), \ \ \ U \in \mathscr{B}(\mathbb{R}^d)$$
is a Poisson point process with the same intensity measure independent of $(\mathscr{F} _{\tau})$. I was not able however to find a reference to this statement.
In terms of random sets, it corresponds to the strong Markov property of the set $[0;\tau] \times \mathbb{R}^d$, which is not compact.