  Considering two independent Poisson processes, we address the question of
testing equality of their respective intensities. We first propose single tests
whose test statistics are U-statistics based on general kernel functions. The
corresponding critical values are constructed from a non-asymptotic wild
bootstrap approach, leading to level \alpha tests. Various choices for the
kernel functions are possible, including projection, approximation or
reproducing kernels. In this last case, we obtain a parametric rate of testing
for a weak metric defined in the RKHS associated with the considered
reproducing kernel. Then we introduce, in the other cases, an aggregation
procedure, which allows us to import ideas coming from model selection,
thresholding and/or approximation kernels adaptive estimation. The resulting
multiple tests are proved to be of level \alpha, and to satisfy non-asymptotic
oracle type conditions for the classical L2-norm. From these conditions, we
deduce that they are adaptive in the minimax sense over a large variety of
classes of alternatives based on classical and weak Besov bodies in the
univariate case, but also Sobolev and anisotropic Nikol'skii-Besov balls in the
multivariate case.
