相应于随机自相似分形的记忆函数和分数次积分

For a physics system which exhibits memory, if memory is preserved only at points of random self-similar fractals, we define random memory functions and give the connection between the expectation of flux and the fractional integral. In particular, when memory sets degenerate to Cantor type fractals or non-random self-similar fractals our results coincide with that of Nigmatullin and Ren et al. .     

Stochastic model for ultraslow diffusion

Ultraslow diffusion is a physical model in which a plume of diffusing particles spreads at a logarithmic rate. Governing partial differential equations for ultraslow diffusion involve fractional time derivatives whose order is distributed over the interval from zero to one. This paper develops the stochastic foundations for ultraslow diffusion based on random walks with a random waiting time between jumps whose probability tail falls off at a logarithmic rate. Scaling limits of these random walks are subordinated random processes whose density functions solve the ultraslow diffusion equation. Along the way, we also show that the density function of any stable subordinator solves an integral equation (5.15) that can be used to efficiently compute this function.     

On one stochastic model that leads to a stable distribution

We consider an integral equation describing the contagion phenomenon, in particular, the equation of the state of a hereditarily elastic body, and interpret this equation as a stochastic model in which the Rabotnov exponent of fractional order plays the role of density of probability of random delay time. We invesgigate the approximation of the distribution for sums of values with a given density to the stable distribution law and establish the principal characteristics of the corresponding renewal process. Ukrainian Mining Academy, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 11, pp. 1572–1579, November, 1997.     

Free continuous actions on zero-dimensional spaces

We show that every countably infinite group admits a free, continuous action on the Cantor set having an invariant probability measure. We also show that every countably infinite group admits a free, continuous action on a non-homogeneous compact metric space and the action is minimal (that is to say, every orbit is dense). In answer to a question posed by Giordano, Putnam and Skau, we establish that there is a continuous, minimal action of a countably infinite group on the Cantor set such that no free continuous action of any group gives rise to the same equivalence relation.     

Fractional integral and its physical interpretation

A relationship is established between Cantor's fractal set (Cantor's bars) and a fractional integral. The fractal dimension of the Cantor set is equal to the fractional exponent of the integral. It follows from analysis of the results that equations in fractional derivatives describe the evolution of physical systems with loss, the fractional exponent of the derivative being a measure of the fraction of the states of the system that are preserved during evolution timet. Such systems can be classified as systems with residual memory, and they occupy an intermediate position between systems with complete memory, on the one hand, and Markov systems, on the other. The use of such equations to describe transport and relaxation processes is discussed. Some generalizations that extent the domain of applicability of the fractional derivative concept are obtained.Kazan State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 354–368, March, 1992.     

On the non-differentiable exact solutions of the (2 + 1)-dimensional local fractional breaking soliton equation on Cantor sets

In this article, a new (2 + 1)-dimensional local fractional breaking soliton equation is derived with the local fractional derivative. Applying the traveling wave transform of the non-differentiable type, the (2 + 1)-dimensional local fractional breaking soliton equation is converted into a nonlinear local fractional ordinary differential equation. By defining a set of elementary functions on Cantor sets, a novel analytical technique namely the Mittag–Leffler function-based method is employed for the first time ever to construct the exact solutions. The solutions on the Cantor sets are presented via the 3-D contours. It reveals that the proposed method is effective and powerful and is expected to give some inspiration for the study of the local fractional PDEs.     

Testing long memory based on a discretely observed process

In this paper we consider the problem of testing long memory for a continuous time process based on high frequency data. We provide two test statistics to distinguish between a semimartingale and a fractional integral process with jumps, where the integral is driven by a fractional Brownian motion with long memory. The small–sample performances of the statistics are evidenced by means of simulation studies. The real data analysis shows that the fractional integral process with jumps can capture the long memory of some financial data.     

GENERALIZED RIEMANN INTEGRAL ON FRACTAL SETS

The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In this paper, by slight modifying the definition of the Henstock integral which was introduced by Jaroslav Kurzweil and Ralph Henstock, we present a new definition of integral on fractal sets. Furthermore, its integrability has been discussed, and the relationship between differentiation and integral is also established. As an example, the integral of Cantor function on Cantor set is calculated.     

On comparability of integral forms

We prove comparability of certain homogeneous anisotropic integral forms. As a consequence we obtain a Hardy type inequality generalising that for the fractional Laplacian. We give an application to anisotropic censored stable processes in Lipschitz domains.     

Space-time fractional Schrödinger equation with time-independent potentials

We develop a space-time fractional Schrödinger equation containing Caputo fractional derivative and the quantum Riesz fractional operator from a space fractional Schrödinger equation in this paper. By use of the new equation we study the time evolution behaviors of the space-time fractional quantum system in the time-independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively. The space-time fractional Schrödinger equation with time-independent potentials is divided into a space equation and a time one. A general solution, which is composed of oscillatory terms and decay ones, is obtained. We investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative, but also on the sign (positive or negative) of the eigenvalues of the space equation. We also find that the limit value of the total probability can be greater or less than one, which means the space-time fractional Schrödinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials. Additionally, the non-Markovian time evolution laws of the space-time fractional quantum system are discussed. The formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space-time fractional quantum system. We also get a Mittag-Leffler type of time evolution operator of wave functions and then establish a Heisenberg equation containing fractional operators.     

Universally measure-preserving homeomorphisms of cantor minimal systems

In 1995, T. Giordano, I. Putnam, and C. Skau [GPS] made a significant breakthrough in Cantor minimal system theory. They completely classified Cantor minimal systems in the sense of topological orbit equivalence by using C*-algebra and homological algebra techniques. Since then, a dynamical proof of their theorem has been conjectured. Such a proof is presented in this paper. We establish orbit equivalence theory based on a finite coordinate change relation arising from an ordered Bratteli diagram, which is known from [HK] in the finitary case of ergodic probability measure-preserving transformations. We obtain the Orbital Extension Theorem. This theorem is considered a topological version of the Copying Lemma of Y. Katznelson and B. Weiss [KW], which has played an important role in measured orbit equivalence theory.     

Stability of quantization dimension and quantization for homogeneous Cantor measures

We effect a stabilization formalism for dimensions of measures and discuss the stability of upper and lower quantization dimension. For instance, we show for a Borel probability measure with compact support that its stabilized upper quantization dimension coincides with its packing dimension and that the upper quantization dimension is finitely stable but not countably stable. Also, under suitable conditions explicit dimension formulae for the quantization dimension of homogeneous Cantor measures are provided. This allows us to construct examples showing that the lower quantization dimension is not even finitely stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integral foliated simplicial volume of aspherical manifolds

Simplicial volumes measure the complexity of fundamental cycles of manifolds. In this article, we consider the relation between the simplicial volume and two of its variants — the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space.     

Convergence Analysis of a Block-by-Block Method for Fractional Differential Equations

The block-by-block method, proposed by Linz for a kind of Volterra integral equations with nonsingular kernels, and extended by Kumar and Agrawal to a class of initial value problems of fractional differential equations (FDEs) with Caputo derivatives, is an efficient and stable scheme. We analytically prove and numerically verify that this method is convergent with order at least 3 for any fractional order index $\alpha>0$.     

Dilated Fractional Stable Motions

Dilated fractional stable motions are stable, self-similar, stationary increments random processes which are associated with dissipative flows. Self-similarity implies that their finite-dimensional distributions are invariant under scaling. In the Gaussian case, when the stability exponent equals 2, dilated fractional stable motions reduce to fractional Brownian motion. We suppose here that the stability exponent is less than 2. This implies that the dilated fractional stable motions have infinite variance and hence they cannot be characterised by a covariance function. These dilated fractional stable motions are defined through an integral representation involving a nonrandom kernel. This kernel plays a fundamental role. In this work, we study the space of kernels for which the dilated processes are well-defined, indicate connections to Sobolev spaces, discuss uniqueness questions and relate dilated fractional stable motions to other self-similar processes. We show that a number of processes that have been obtained in the literature, are in fact dilated fractional stable motions, for example, the telecom process obtained as limit of renewal reward processes, the Takenaka processes and the so-called random wavelet expansion processes.     

A New Approach to Fractional Brownian Motion of Order n Via Random Walk in the Complex Plane

By using a very simple model of random walk defined on the roots of the unity in the complex plane, one can obtain the model of fractional brownian motion of order n which has been previously introduced in the form of rotating Gaussian white noise. This definition of fractional Brownian motion of order n as the limit of complex random walk, provides new insights in its genuine practical meaning, and in the derivation of most of the related theoretical results. Itôs stochastic calculus can be extended in a straightforward manner to the path integral so generated in the complex plane. The corresponding probability distribution is stable in Levys sense, a Lindebergs like central limit theorem is stated, together with a Feyman–Kacs formula and a Dinkins formula. Then one exhibits the relation between the Hausdorffs dimension and the pattern entropy of the process. The probabilistic approach here is different from Hochbergs and Mandelbrots. Like Saintys, it uses the complex roots of the unity, but it is much more straightforward and simple, and it is the only one which provides results which are fully consistent with the so-called Kramers–Moyal expansion.     

Dirichlet forms and associated heat kernels on the Cantor set induced by random walks on trees

Transient random walk on a tree induces a Dirichlet form on its Martin boundary, which is the Cantor set. The procedure of the inducement is analogous to that of the Douglas integral on S¹ associated with the Brownian motion on the unit disk. In this paper, those Dirichlet forms on the Cantor set induced by random walks on trees are investigated. Explicit expressions of the hitting distribution (harmonic measure) ν and the induced Dirichlet form on the Cantor set are given in terms of the effective resistances. An intrinsic metric on the Cantor set associated with the random walk is constructed. Under the volume doubling property of ν with respect to the intrinsic metric, asymptotic behaviors of the heat kernel, the jump kernel and moments of displacements of the process associated with the induced Dirichlet form are obtained. Furthermore, relation to the noncommutative Riemannian geometry is discussed.     

Well–posedness for an Abstract Semilinear Integro–differential Fractional Problem

We consider an abstract second order semilinear integrodifferential equation involving fractional time derivatives of order between 1 and 2. Well–posedness is established under appropriate conditions on the initial data and the nonlinearity which is itself a fractional integral. These conditions will determine the exact underlying space where to look for solutions.     

离散分数次积分的加权有界性

本文通过引入Z上的逆H?lder类RHr(Z)(r∈(1,∞))并建立它与Z上的Muckenhoupt权空间Aq(Z)(q∈[1,∞))之间的关系,再借助伪差分算子(包含离散Fourier乘子)的有界性与它的积分核的估计之间的关系,获得离散分数次积分在加权离散(弱型)Lebesgue空间lwp(Z)和lwp,∞(Z)(p∈[1,∞))上的有界性.     

Weighted Weak-type Estimates for Multilinear Commutators of Fractional Integrals on Spaces of Homogeneous Type

We obtain weighted distributional inequalities for multilinear commutators of the fractional integral on spaces of homogeneous type, The techniques developed in this work involve the behavior of some fractional maximal functions. In relation to these operators, as a main tool, we prove a weighted weak type boundedness result, which is interesting in itself.