Kolmogorov-Feller equation

The space-fractional equation (3.91) can be derived from the Kolmogorov-Feller equation (3.74) by using the assumption that the random jump Z has a Levy-stable PDF Wg,(z), symmetric with respect to zero, with power-law tails as z -> oo. There is no general explicit form for u (z), but the characteristic function of iv z), the structure function, is... 

In this section we show how to obtain subdiffusive transport by using the idea of inverse subordination [278, 371]. Assume that the density 0 obeys the Kolmogorov-Feller equation... 

UX = const, w(z) does not depend on x, and v(x) = 0, we obtain the Kolmogorov-Feller equation (3.74), for which the underlying microscopic random movement is a compound Poisson process. 

By using the same cmicepts, a very large niun-ber of other problems may be solved. Such an example the probability density function of a random variable may be obtained with the same technique here used for representing cross-correlations in terms of FSMs. It follows that Fokker-Planck equation, Kolmogorov-Feller equation, Einstein-Smoluchowski equation, and path integral solution (Cottone et al. 2008) may be solved in terms of FSM. Moreover, wavelet transform and classical or fractional differential equations may be easily solved by using fractional calculus and Mellin transform in complex domain. 

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