  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
$\tilde{f}_{n,\gamma}$ 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 $\tilde{f}_{n,\gamma}$ on Besov spaces ${\cal
B}^{\ensuremath \alpha}_{p,q}$ are established. Under mild assumptions, we
prove that $$\sup_{f\in B^{\ensuremath \alpha}_{p,q}\cap \ensuremath \mathbb
{L}_{\infty}} \ensuremath \mathbb {E}(\ensuremath | | \tilde{f}_{n,\gamma}-f|
|_2^2)\leq C(\frac{\log n}{n})^{\frac{\ensuremath \alpha}{\ensuremath
\alpha+{1/2}+({1/2}-\frac{1}{p})_+}}$$ and the lower bound of the minimax risk
for ${\cal B}^{\ensuremath \alpha}_{p,q}\cap \ensuremath \mathbb {L}_{\infty}$
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 ${\cal B}^{\ensuremath \alpha}_{p,q}$ 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, $\tilde{f}_{n,\gamma}$
is adaptive minimax up to a logarithmic term.
