  We consider a Markovian jumping process which is defined in terms of the
jump-size distribution and the waiting-time distribution with a
position-dependent frequency, in the diffusion limit. We assume the power-law
form for the frequency. For small steps, we derive the Fokker-Planck equation
and show the presence of the normal diffusion, subdiffusion and superdiffusion.
For the L\'evy distribution of the step-size, we construct a fractional
equation, which possesses a variable coefficient, and solve it in the diffusion
limit. Then we calculate fractional moments and define fractional diffusion
coefficient as a natural extension to the cases with the divergent variance. We
also solve the master equation numerically and demonstrate that there are
deviations from the L\'evy stable distribution for large wave numbers.
