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36
The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics
, 1996
"... We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers b ..."
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Cited by 18 (1 self)
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We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, socalled generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p conditions, singular and fractional diffusion operators. Subsequently, we analyz...
Fractional Calculus and Stable Probability Distributions
, 1998
"... Fractional calculus allows one to generalize the linear (one dimensional) diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of a fractional order. The fundamental solutions of these generalized diffusion equations are shown to provide ..."
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Cited by 13 (2 self)
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Fractional calculus allows one to generalize the linear (one dimensional) diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of a fractional order. The fundamental solutions of these generalized diffusion equations are shown to provide certain probability density functions, in space or time, which are related to the relevant class of stable distributions. For the space fractional diffusion a randomwalk model is also proposed. Keywords  Fractional calculus, diffusion equation, stable distributions, randomwalk. 1. Introduction The purpose of this note is to outline the role of fractional calculus in generating stable probability distributions through generalized diffusion equations of fractional order. For the standard diffusion equation it is well known that the fundamental solution of the Cauchy problem provides the spatial probability density function (pdf) for the Gaussian or normal distribution, whose variance...
Numerical Simulations of Anomalous Diffusion
, 2003
"... In this paper we present numerical methods  finite differences and finite elements  for solution of partial differential equation of fractional order in time for onedimensional space. This equation describes anomalous diffusion which is a phenomenon connected with the interactions within the comp ..."
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Cited by 7 (4 self)
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In this paper we present numerical methods  finite differences and finite elements  for solution of partial differential equation of fractional order in time for onedimensional space. This equation describes anomalous diffusion which is a phenomenon connected with the interactions within the complex and nonhomogeneous background. In order to consider physical initialvalue conditions we use fractional derivative in the Caputo sense. In numerical analysis the boundary conditions of first kind are accounted and in the final part of this paper the result of simulations are presented.
Generalized Dirichlet forms and associated Markov processes
 C.R. Acad. Paris
, 1994
"... We prove that for a certain class of bilinear forms satisfying some regularity conditions which include quasiregular Dirichlet forms (cf. [3]) and time dependent Dirichlet forms (cf. [5]) as particular cases there exists an associated strong Markov process having nice sample path properties. These ..."
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Cited by 6 (2 self)
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We prove that for a certain class of bilinear forms satisfying some regularity conditions which include quasiregular Dirichlet forms (cf. [3]) and time dependent Dirichlet forms (cf. [5]) as particular cases there exists an associated strong Markov process having nice sample path properties. These forms, called generalized Dirichlet forms, are the sum of a coercive part and a perturbation (e.g. the time derivative in the time dependent case), so that in general neither the sector condition is fulfilled by the sum nor is the associated L 2 semigroup analytic. A wide variety of new examples can be treated in this extended framework of Dirichlet forms including fractional diffusion operators and transformations of time dependent Dirichlet forms by ffexcessive functions h (htransformations). Formes de Dirichlet g'en'eralis'ees et processus de Markov associ'es R'esum'e  Nous construisons des processus standard sp'eciaux associ'es `a certaines formes bilin'eaires qui satisfont `a quel...
Fractional Oscillations And MittagLeffler Functions
, 1996
"... The fractional oscillation equation is obtained from the classical equation for linear oscillations by replacing the secondorder time derivative by a fractional derivative of order ff with 1 ! ff ! 2 : Using the method of the Laplace transform, it is shown that the fundamental solutions can be expr ..."
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Cited by 6 (2 self)
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The fractional oscillation equation is obtained from the classical equation for linear oscillations by replacing the secondorder time derivative by a fractional derivative of order ff with 1 ! ff ! 2 : Using the method of the Laplace transform, it is shown that the fundamental solutions can be expressed in terms of MittagLeffler functions, and exhibit a finite number of damped oscillations with an algebraic decay. For completeness we also discuss both the cases 0 ! ff ! 1 (fractional relaxation) and 2 ! ff 3 (growing oscillations), showing the key role of the MittagLeffler functions. 1991 Mathematics Subject Classification: 26A33, 33E20, 33E30, 45E10, 45J05, 70J99. Key Words: Fractional differential equations, fractional calculus, MittagLeffler functions. 1. Introduction For real ff ? 0 (later only for 0 ! ff 3) we consider the fractional differential equation D ff / u(t) \Gamma m\Gamma1 X k=0 t k k! u (k) (0 + ) ! = \Gammau(t) + q(t) ; t ? 0 ; (1:1) where q(t...
From power laws to fractional diffusion: the direct way
 Vietnam Journal of Mathematics 32 SI
, 2004
"... Starting from the model of continuous time random walk (Montroll and Weiss 1965) that can also be considered as a compound renewal process we focus our interest on random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with expon ..."
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Cited by 6 (6 self)
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Starting from the model of continuous time random walk (Montroll and Weiss 1965) that can also be considered as a compound renewal process we focus our interest on random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. By stating the relevant lemmata (of Tauber type) for the distribution functions we need not distinguish between continuous and discrete space and time. We will see that by a wellscaled passage to the diffusion limit diffusion processes fractional in time as well as in space are obtained. The corresponding equation of evolution is a linear partial pseudodifferential equation with fractional derivatives in time and in space, the orders being equal to the above exponents. Such processes are enjoying increasing popularity in applications in physics, chemistry, finance and other fields, and their behaviour can be well approximated and visualized by simulation via various types of random walks. For their explicit solutions there are available integral representations that allow to investigate their detailed structure. For ease of presentation we restrict attention to the spatially onedimensional symmetric situation.
Probability Distributions Generated By Fractional Diffusion Equations
, 1999
"... . Fractional calculus allows one to generalize the linear onedimensional diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of these generalized diffusion equations are shown to provide probab ..."
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Cited by 6 (3 self)
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. Fractional calculus allows one to generalize the linear onedimensional diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of these generalized diffusion equations are shown to provide probability density functions evolving in time or varying in space which are related to the special class of stable distributions. This property is a noteworthy generalization of what happens in the case of the standard diffusion equation and can be relevant in treating financial and economical problems where stable probability distributions are known to play a key role. 1. Introduction NonGaussian probability distributions are becoming more common as data models, especially in economics, where large fluctuations are expected. In fact, probability distributions with heavy tails are often met in economics and finance, which suggests enlarging the arsenal of possible stochastic models by nonGauss...
Dirichlet Forms And Markov Processes: A Generalized Framework Including Both Elliptic And Parabolic Cases
"... We extend the framework of classical Dirichlet forms to a class of bilinear forms, called generalized Dirichlet forms, which are the sum of a coercive part and a linear unbounded operator as a perturbation. The class of generalized Dirichlet forms, in particular, includes symmetric and coercive Dir ..."
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Cited by 5 (3 self)
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We extend the framework of classical Dirichlet forms to a class of bilinear forms, called generalized Dirichlet forms, which are the sum of a coercive part and a linear unbounded operator as a perturbation. The class of generalized Dirichlet forms, in particular, includes symmetric and coercive Dirichlet forms (cf. [Fu2], [M/R]) as well as time dependent Dirichlet forms (cf. [O1]) as special cases and also many new examples. Among these are, e.g. transformations of time dependent Dirichlet forms by ffexcessive functions h (htransformations), Dirichlet forms with time dependent linear drift and fractional diffusion operators. One of the main results is that we identify an analytic property of these forms which ensures the existence of associated strong Markov processes with nice sample path properties, and give an explicit construction for such processes. This construction extends previous constructions of the processes in the elliptic and the parabolic cases, is, in particular, c...
Distributed Order Calculus and Equations of Ultraslow Diffusion
"... 1 Abstract We consider equations of the formi D(u)uj (t, x) \Delta u(t, x) = f (t, x), t> 0, x 2 Rn, where D(u) is a distributed order derivative, that is D(u)'(t) = ..."
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Cited by 2 (1 self)
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1 Abstract We consider equations of the formi D(u)uj (t, x) \Delta u(t, x) = f (t, x), t> 0, x 2 Rn, where D(u) is a distributed order derivative, that is D(u)'(t) =