  L\'{e}vy flights can be described using a Fokker-Planck equation which
involves a fractional derivative operator in the position co-ordinate. Such an
operator has its natural expression in the Fourier domain. Starting with this,
we show that the solution of the equation can be written as a Hamiltonian path
integral. Though this has been realized in the literature, the method has not
found applications as the path integral appears difficult to evaluate. We show
that a method in which one integrates over the position co-ordinates first,
after which integration is performed over the momentum co-ordinates, can be
used to evaluate several path integrals that are of interest. Using this, we
evaluate the propagators for (a) free particle (b) particle subjected to a
linear potential and (c) harmonic potential. In all the three cases, we have
obtained results for both overdamped and underdamped cases.
