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The work of T. Srokowski (markovian-levy-diffusion-2006 non-markovian-levy-diffusion-srokowski) gives a picture in physics of Markovian and non-Markovian diffusion with Levy distributed jump sizes.  The Hurst exponent distribution empirically observed for a large number of stocks over many years tells us that the underlying process for returns spans both the subdiffusive and superdiffusive cases.  The mapping to the physicists concepts is provided by $ ~ t^{2H}$ where $H$ is the empirically determined Hurst exponent by whatever method.  The following histogram of Hurst exponent distributions uses the original Rescaled Range Analysis used by Hurst and Mandelbrot.
The Fokker-Planck equation for probability density function $p(t,x)$ for Markovian Levy case is $(d/dt + A_1)p(t,x) = 0$ and non-Markovian case is $(d/dt + A_2)p(t,x) = 0$ where $A_1 = K^\alpha \frac{\partial^\alpha}{\partial |x|^\alpha} D(x)$ and $A_2 = (d/dt)^\gamma A_1$.  Fox $H$-functions appears in the Green’s functions in Markovian case.