We take a wavelet based approach to the analysis of point processes and the estimation of the first order intensity under a continuous time setting. A multiresolution analysis of a point process is formulated which motivates the definition of homogeneity at different scales of resolution, termed $J$-th level homogeneity. Further to this, the activity in a point processes' first order behavior at different scales of resolution is also defined and termed $L$-th level innovation. Likelihood ratio tests for both these properties are proposed with asymptotic distributions provided, even when only a single realization of the point process is observed. The test for $L$-th level innovation forms the basis for a collection of statistical strategies for thresholding coefficients in a wavelet based estimator of the intensity function. These thresholding strategies are shown to outperform the existing local hard thresholding strategy on a range of simulation scenarios.