By Meerschaert M.M., Sikorskii A.
This monograph develops the fundamental thought of fractional calculus and anomalous diffusion, from the perspective of chance. The reader will see how fractional calculus and anomalous diffusion will be understood at a deep and intuitive point, utilizing rules from likelihood. The publication covers simple restrict theorems for random variables and random vectors with heavy tails. Heavy tails are utilized in finance, coverage, physics, geophysics, telephone biology, ecology, drugs, and computing device engineering
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Additional info for Stochastic Models for Fractional Calculus
Sample text
The characteristic function μ(k) ˆ = E eikY = eikx μ(dx) = eikx F (dx) = eikx f (x) dx = fˆ(−k) is related to the Fourier transform (FT) by an obvious change of sign. 11) for moments. 14) for derivatives. See the details and the end of this section for more information. We say that (the distribution of) Y is infinitely divisible if Y X1 + · · · + Xn for every positive integer n, where (Xn ) are independent and identically distributed (iid) random variables. If Xn μn , then we also have μ(k) ˆ = E[eikY ] = E[eik(X1 +···+Xn ) ] = E[eikX1 ] · · · E[eikXn ) ] = μˆ n (k)n since X1 , .
If the pdf f (x) = F (x) exists, then we can define the probability measure b μ(a, b] = P[a < Y ≤ b] = a f (x) dx, 54 Chapter 3 Stable Limit Distributions and the characteristic function μ(k) ˆ = eikx f (x) dx. , if P[Y = xk ] > 0 for some real numbers xk , then F (xk ) > F (xk −) and the cumulative distribution function is not continuous, so it is certainly not differentiable. Then the pdf cannot exist at every x ∈ R. In this case, the characteristic function μ(k) ˆ = E eikY = eikx μ(dx) = eikx F (dx) is defined using the Lebesgue integral with respect to the probability measure μ, or equivalently, the Lebesgue-Stieltjes integral with respect to the cumulative distribution function F (x).
182]. 8): ∂p ∂2p ∂p ∂ −D =D 2 =− ∂t ∂x ∂x ∂x assuming D is a constant independent of x. 9) can be understood in terms of the Grünwald formula. 1. The fractional Fick’s Law for the flux allows particles to jump into the box at location x from a box at location x − jΔx. The proportion of particles that make a jump this long drops off as a power of the separation distance. See Schumer et al. [182] for more details. 10) which leads to the same fractional diffusion equation ∂p ∂p ∂ α−1 ∂αp = − α−1 −D = D α, ∂t ∂x ∂x ∂x see Meerschaert, Mortensen and Wheatcraft [129] for additional details.