  We consider the Cauchy problem for a pseudo-differential operator which has a
translation-invariant and analytic symbol. For a certain set of initial
conditions, a formal solution is obtained by a perturbative expansion. The
series so obtained can be re-expressed in terms of generalized Feynman graphs
and Feynman rules. The logarithm of the solution can then be represented by a
series containing only the connected Feynman graphs. Under some conditions, it
is shown that the formal solution uniquely determines the real solution by
means of Borel transforms. The formalism is then applied to probabilistic
L\'{e}vy distributions. Here, the Gaussian part of such a distribution is
re-interpreted as a initial condition and a large diffusion expansion for
L\'{e}vy densities is obtained. It is outlined how this expansion can be used
in statistical problems that involve L\'{e}vy distributions.
