  The L\'evy, jumping process, defined in terms of the jumping size
distribution and the waiting time distribution, is considered. The jumping rate
depends on the process value. The fractional diffusion equation, which contains
the variable diffusion coefficient, is solved in the diffusion limit. That
solution resolves itself to the stretched Gaussian when the order parameter
$\mu\to2$. The truncation of the L\'evy flights, in the exponential and
power-law form, is introduced and the corresponding random walk process is
simulated by the Monte Carlo method. The stretched Gaussian tails are found in
both cases. The time which is needed to reach the limiting distribution
strongly depends on the jumping rate parameter. When the cutoff function falls
slowly, the tail of the distribution appears to be algebraic.

Statistical Mechanics