Statistical dynamics of stable processes

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Abstract
We consider the Markovian evolution in phase space of a distribution of particles subject both to a regular external field and to stochastic forces whose transition probabilities are Lévy stable densities. An integro-differential equation is derived which naturally generalizes the Fokker-Planck equation much as the stable densities generalize the normal. Its compact expression employs a new notation of vector fractional derivatives. Solutions are obtained for equilibrium conditions as well as for field-free and harmonically bound particle forces. The real-space Smoluchowski-equation limit of the Fokker-Planck equation is also generalized. It is suggested that the general evolution equation and stable densities may describe statistically the strange or fractal behavior of turbulent systems in the neighborhood of their critical points in phase space, with generally noninteger-order derivative laws governing the turbulent diffusion.

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