  Consider a random process s solution of the stochastic partial differential
equation Ls = w with L a homogeneous operator and w a multidimensional L\'evy
white noise. In this paper, we study the asymptotic effect of zooming in or
zooming out of the process s. More precisely, we give sufficient conditions on
L and w so that the rescaled versions of s converges in law to a self-similar
process of order H at coarse scales and at fine scales. The parameter H depends
on the homogeneity order of the operator L and the Blumenthal-Getoor indices
associated to the L\'evy white noise w. Finally, we apply our general results
to several notorious classes of random processes and random fields.
