  We study the long-time behavior of underdamped Brownian particle moving
through a viscous medium and in a systematic potential, when it is subjected to
a space-dependent high-frequency periodic force. When the frequency is very
large, much larger than all other relevant system-frequencies, there is a
Kapitsa time-window wherein the effect of frequency dependent forcing can be
replaced by a static effective potential. Our new analysis includes the case
when the forcing, in addition to being frequency-dependent, is space-dependent
as well. The results of the Kapitsa analysis then lead to additional
contributions to the effective potential. These are applied to the numerical
calculation of the diffusion coefficient (D) for a Brownian particle moving in
a periodic potential. Presented are numerical results, which are in excellent
agreement with theoretical predictions and which indicate a significant
enhancement of D due to the space-dependent forcing terms. In addition we study
the transport property (current) of underdamped Brownian particles in a ratchet
potential.
