  The nonlinear oscillator model is useful to basically understand the most
important properties of nonlinear optical processes. It has been shown to give
the correct asymptotic behaviour and to provide the general feature of harmonic
generation to all orders, in particular dispersion relations and sum rules. We
investigate the properties of Pump and Probe processes using this model, and
study those cases where general theorems based on the holomorphic character of
the Kubo response function cannot be applied. We show that it is possible to
derive new sum rules and new Kramers-Kronig relations for the two lowest
moments of the real and imaginary part of the third order susceptibility and
that new specific contributions become relevant as the intensity of the probe
increases. Since the analitic properties of the susceptibility functions depend
only upon the time causality of the system we are confident that these results
are not model depemdent and therefore have a general validity, provided one
substitutes to the equilibrium values of the potential derivatives the density
matrix expectation values of the corresponding operators.
