The purpose of this paper is to estimate the intensity of a Poisson process $N$ by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of $N$ with respect to $ndx$ where $n$ is a fixed parameter, is assumed to be non-compactly supported. The estimator $\tilde{f}_{n,γ}$ based on random thresholds is proved to achieve the same performance as the oracle estimator up to a possible logarithmic term. Then, minimax properties of $\tilde{f}_{n,γ}$ on Besov spaces ${\cal B}^{\ensuremath α}_{p,q}$ are established. Under mild assumptions, we prove that $$\sup_{f\in B^{\ensuremath α}_{p,q}\cap \ensuremath \mathbb {L}_{\infty}} \ensuremath \mathbb {E}(\ensuremath | | \tilde{f}_{n,γ}-f| |_2^2)\leq C(\frac{\log n}{n})^{\frac{\ensuremath α}{\ensuremath α+{1/2}+({1/2}-\frac{1}{p})_+}}$$ and the lower bound of the minimax risk for ${\cal B}^{\ensuremath α}_{p,q}\cap \ensuremath \mathbb {L}_{\infty}$ coincides with the previous upper bound up to the logarithmic term. This new result has two consequences. First, it establishes that the minimax rate of Besov spaces ${\cal B}^{\ensuremath α}_{p,q}$ with $p\leq 2$ when non compactly supported functions are considered is the same as for compactly supported functions up to a logarithmic term. When $p>2$, the rate exponent, which depends on $p$, deteriorates when $p$ increases, which means that the support plays a harmful role in this case. Furthermore, $\tilde{f}_{n,γ}$ is adaptive minimax up to a logarithmic term.