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The L\'evy, jumping process, defined in terms of the jumping sizedistribution and the waiting time distribution, is considered. The jumping ratedepends on the process value. The fractional diffusion equation, which containsthe variable diffusion coefficient, is solved in the diffusion limit. Thatsolution resolves itself to the stretched Gaussian when the order parameter$\mu\to2$. The truncation of the L\'evy flights, in the exponential andpower-law form, is introduced and the corresponding random walk process issimulated by the Monte Carlo method. The stretched Gaussian tails are found inboth cases. The time which is needed to reach the limiting distributionstrongly depends on the jumping rate parameter. When the cutoff function fallsslowly, the tail of the distribution appears to be algebraic.

Stretched-Gaussian asymptotics of the truncated L\'evy flights for the diffusion in nonhomogeneous media