The standard central limit theorem plays a fundamental role in Boltzmann-Gibbs statistical mechanics. This important physical theory has been generalized [1] in 1988 by using the entropy S q =The theory which emerges is usually referred to as nonextensive statistical mechanics and recovers the standard theory for q = 1. During the last two decades, this q-generalized statistical mechanics has been successfully applied to a considerable amount of physically interesting complex phenomena. A conjecture[2] and numerical indications available in the literature have been, for a few years, suggesting the possibility of q-versions of the standard central limit theorem by allowing the random variables that are being summed to be strongly correlated in some special manner, the case q = 1 corresponding to standard probabilistic independence. This is what we prove in the present paper for 1 ≤ q < 3. The attractor, in the usual sense of a central limit theorem, is given by a distribution of the form p(x) = C q [1 − (1 − q)βx 2 ] 1/(1−q) with β > 0, and normalizing constant C q . These distributions, sometimes referred to as q-Gaussians, are known to make, under appropriate constraints, extremal the functional S q (in its continuous version). Their q = 1 and q = 2 particular cases recover respectively Gaussian and Cauchy distributions. Mathematics Subject Classification (2000). Primary 60F05; Secondary 60E07, 60E10, 82Cxx.

The alpha-stable distributions introduced by Lévy play an important role in probabilistic theoretical studies and their various applications, e.g., in statistical physics, life sciences, and economics. In the present paper we study sequences of long-range dependent random variables whose distributions have asymptotic power-law decay, and which are called (q,alpha)-stable distributions. These sequences are generalizations of independent and identically distributed alpha-stable distributions and have not been previously studied. Long-range dependent (q,alpha)-stable distributions might arise in the description of anomalous processes in nonextensive statistical mechanics, cell biology, finance. The parameter q controls dependence. If q=1 then they are classical independent and identically distributed with alpha-stable Lévy distributions. In the present paper we establish basic properties of (q,alpha)-stable distributions and generalize the result of Umarov et al. [Milan J. Math. 76, 307 (2008)], where the particular case alpha=2,q[1,3) was considered, to the whole range of stability and nonextensivity parameters alpha(0,2] and q[1,3), respectively. We also discuss possible further extensions of the results that we obtain and formulate some conjectures.

Fractional order differential equations are an efficient tool to model various processes arising in science and engineering. Fractional models adequately reflect subtle internal properties, such as memory or hereditary properties, of complex processes that the classical integer order models neglect. In this chapter we will discuss the theoretical background of fractional modeling, that is the fractional calculus, including recent developments -distributed and variable fractional order differential operators.Fractional order derivatives interpolate integer order derivatives to real (not necessarily fractional) or complex order derivatives. There are different types of fractional derivatives not always equivalent. The first attempt to develop the fractional calculus systematically was taken by Liouville (1832) and Riemann (1847) in the first half of the nineteenth century, even though discussions on non-integer order derivatives had been started long ago. 1 In the 1870s Letnikov and Grünwald independently used an approach for the definition of the fractional order derivative and integral different from that of Riemann and Liouville. The Cauchy problem for fractional order differential equations with the Riemann-Liouville derivative is not well posed (Section 3.3), that is the Cauchy problem in this case is unphysical. In the 1960s Caputo and Djrbashian introduced independently, so-called, a regularization of the Riemann-Liouville fractional derivative, which was later named a fractional derivative in the sense of Caputo-Djrbashian (Section 3.5). The usefulness of the Caputo-Djrbashian derivative is that the Cauchy problem for fractional order differential equations with the Caputo-Djrbashian derivative is well posed. In the operators language one can write the latter in the form DJ = I, where I is the identity operator, which means that the operator D is a left inverse to the operator J. One can easily check that D is not a right inverse to J, since, according to the same fundamental theorem of calculus, for any differentiable function f the equality JD f (t) = f (t) − f (0) holds. The similar relations are true by induction for operators D n and J n , where D n = d n dt n , "n-th derivative," and J n is the n-fold integration operator. Namely,andThus D n is the left inverse to J n , and is the right inverse to J n in the class of functions satisfying additional conditions: f (k) (0) = 0, k = 0,..., n − 1. These relations between "differentiation" and "integration" operators valid for n = 1, 2, . . . , form the basis for the definitions of fractional derivatives in the sense of RiemannLiouville and Caputo-Djrbashian, as soon as the fractional order integration operator is defined. Fractional order integration operator 123 Fractional order integration operatorIn this section we introduce the fractional order integration operator of order α > 0 (ℜ(α) > 0). One can verify (by changing order of integration) that the n-fold integration operatorTaking into account the relationship Γ (n) = (n − 1)!, whereis the Euler's ...

It is known that the transition probabilities of a solution to a classical Itô stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coefficients determined by the corresponding SDE. Timefractional Kolmogorov type equations are used to model complex processes in many fields. However, the class of SDEs that is associated with these equations is unknown except in a few special cases. The present paper shows that in the cases of either time-fractional order or more general time-distributed order differential equations, the associated class of SDEs can be described within the framework of SDEs driven by semimartingales. These semimartingales are time-changed Lévy processes where the independent time-change is given respectively by the inverse of a single or mixture of independent stable subordinators. Examples are provided, including a fractional analogue of the Feynman-Kac formula.

In this paper Fokker-Planck-Kolmogorov type equations associated with stochastic differential equations driven by a time-changed fractional Brownian motion are derived. Two equivalent forms are suggested. The time-change process considered is either the first hitting time process for a stable subordinator or a mixture of stable subordinators. A family of operators arising in the representation of the Fokker-Plank-Kolmogorov equations is shown to have the semigroup property.