  The influence functional method of Feynman and Vernon is used to obtain a
quantum master equation for a Brownian system subjected to a Levy stable random
force. The corresponding classical transport equations for the Wigner function
are then derived, both in the limit of weak and strong friction. These are
fractional extensions of the Klein-Kramers and the Smoluchowski equations. It
is shown that the fractional character acquired by the position in the
Smoluchowski equation follows from the fractional character of the momentum in
the Klein-Kramers equation. Connections among fractional transport equations
recently proposed are clarified.
