  We propose to test the homogeneity of a Poisson process observed on a finite
interval. In this framework, we first provide lower bounds for the uniform
separation rates in $\mathbb{L}^2$ norm over classical Besov bodies and weak
Besov bodies. Surprisingly, the obtained lower bounds over weak Besov bodies
coincide with the minimax estimation rates over such classes. Then we construct
non asymptotic and nonparametric testing procedures that are adaptive in the
sense that they achieve, up to a possible logarithmic factor, the optimal
uniform separation rates over various Besov bodies simultaneously. These
procedures are based on model selection and thresholding methods. We finally
complete our theoretical study with a Monte Carlo evaluation of the power of
our tests under various alternatives.
