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 Lévy 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 the 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 Lévy stable distribution for large wave numbers.

Langevin equation with a multiplicative stochastic force is considered. That force is uncorrelated, it has the Lévy distribution and the power-law intensity. The Fokker-Planck equations, which correspond both to the Itô and Stratonovich interpretation, are presented. They are solved for the case without drift and for the harmonic oscillator potential. The variance is evaluated; it is always infinite for the Itô case whereas for the Stratonovich one it can be finite and rise with time slower that linearly, which indicates subdiffusion. Analytical results are compared with numerical simulations.

The fractional Fokker-Planck equation, which contains a variable diffusion coefficient, is discussed and solved. It corresponds to the Lévy flights in a nonhomogeneous medium. For the case with the linear drift, the solution is stationary in the long-time limit and it represents the Lévy process with a simple scaling. The solution for the drift term in the form lambda sgn(x) possesses two different scales which correspond to the Lévy indexes micro and micro+1 (micro<1) . The former component of the solution prevails at large distances but it diminishes with time for a given x . The fractional moments, as a function of time, are calculated. They rise with time and the rate of this growth increases with lambda .

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The Langevin equation with a multiplicative Lévy white noise is solved. The noise amplitude and the drift coefficient have a power-law form. A validity of ordinary rules of the calculus for the Stratonovich interpretation is discussed. The solution has the algebraic asymptotic form and the variance may assume a finite value for the case of the Stratonovich interpretation. The problem of escaping from a potential well is analyzed numerically; predictions of different interpretations of the stochastic integral are compared.