Linear relaxation processes governed by fractional symmetric kinetic equations

The fractional symmetric Fokker-Planck and Einstein-Smoluchowski kinetic eq uations that describe the evolution of systems influenced by stochastic for ces distributed with stable probability laws are derived. These equations g eneralize the known kinetic equations of the Brownian motion theory and inv olve symmetric fractional derivatives with respect to velocity and space va riables. With the help of these equations, the linear relaxation processes in the force-free case and for the linear oscillator is analytically studie d. For a weakly damped oscillator, a kinetic equation for the distribution in slow variables is obtained. Linear relaxation processes are also studied numerically by solving the corresponding Langevin equations with the sourc e given by a discrete-time approximation to white Levy noise. Numerical and analytical results agree quantitatively. (C) 2000 MAIK "Nauka/Interperiodi ca".