Sabir Umarov

Continuous time random walk models associated with distributed order diffusion equations

Abstract In this paper continuous time and discrete random walk models approximating diffusion processes associated with time-fractional and spacedistributed order differential equations are studied. Stochastic processes associated with the considered equations represent time-changed processes, where the time-change process is the inverse to a Levy’s stable subordinator with the stability index β ∈ (0, 1). In the paper the convergence of modeled continuous time and discrete random walks to time-changed processes associated with distributed order fractional diffusion equations are proved using an analytic method.

Introduction to fractional and pseudo-differential equations with singular symbols

Function Spaces and Distributions.- Pseudo-Differential Operators with Singular Symbols (DOSS).- Fractional Calculus and Fractional Order Operators.- Boundary Value Problems for Pseudo-Differential Equations with Singular Symbols.- Initial and Boundary Value Problems for Fractional Order Differential Equations.- Distributed and Variable Order Differential-Operator Equations.- Fractional Fokker-Planck-Kolmogorov Equations.- Random Walk Approximants of Mixed and Time-Changed Levy Processes.- Complex DOSS and Systems of Complex Differential Equations.- References.

Fractional Fokker-Planck-Kolmogorov equations associated with SDES on a bounded domain

Abstract This paper is devoted to the fractional generalization of the Fokker-Planck equation associated with a nonlinear stochastic differential equation on a bounded domain. The driving process of the stochastic differential equation is a Lévy process subordinated to the inverse of Lévy’s mixed stable subordinators. The Fokker-Planck equation is given through the general Waldenfels operator, while the boundary condition is given through the general Wentcel’s boundary condition. As a fractional operator a distributed order differential operator with a Borel mixing measure is considered. In the paper fractional generalizations of the Fokker-Planck equation are derived and the existence of a unique solution of the corresponding initial-boundary value problems is proved.

Use of the geometric mean as a statistic for the scale of the coupled Gaussian distributions

Abstract The geometric mean is shown to be an appropriate statistic for the scale of a heavy-tailed coupled Gaussian distribution or equivalently the Student’s t distribution. The coupled Gaussian is a member of a family of distributions parameterized by the nonlinear statistical coupling which is the reciprocal of the degree of freedom and is proportional to fluctuations in the inverse scale of the Gaussian. Existing estimators of the scale of the coupled Gaussian have relied on estimates of the full distribution, and they suffer from problems related to outliers in heavy-tailed distributions. In this paper, the scale of a coupled Gaussian is proven to be equal to the product of the generalized mean and the square root of the coupling. From our numerical computations of the scales of coupled Gaussians using the generalized mean of random samples, it is indicated that only samples from a Cauchy distribution (with coupling parameter one) form an unbiased estimate with diminishing variance for large samples. Nevertheless, we also prove that the scale is a function of the geometric mean, the coupling term and a harmonic number. Numerical experiments show that this estimator is unbiased with diminishing variance for large samples for a broad range of coupling values.

Fractional generalizations of Zakai equation and some solution methods

Abstract The paper discusses fractional generalizations of Zakai equations arising in filtering problems. The derivation of the fractional Zakai equation, existence and uniqueness of its solution, as well as some methods of solution to the fractional filtering problem, including fractional version of the particle flow method, are presented.

Pseudo-differential operators with singular symbols (ΨDOSS)

We begin Chapter 2 with simple examples of initial and boundary value problems, solution operators of which have singularities of one or another type in the dual variable. The presence of a singularity often causes a failure of well posedness of the problem in the sense of Hadamard. Let A be a linear differential operator mapping a function space X into another function space F.

Function spaces and distributions

This chapter is devoted to function and distribution spaces. We first recall definitions of some well-known classical function and distribution spaces, simultaneously introducing the terminology and notations used in this book. Then we introduce (see Section 1.10) a new class of test functions and the corresponding space of distributions (generalized functions), which play an important role in the theory of pseudo-differential operators with singular symbols introduced in Chapter 2 By singular symbols we mean, if not otherwise assumed, symbols singular in dual variables.

Fractional order Fokker-Planck-Kolmogorov equations and associated stochastic processes

This chapter discusses the connection between pseudo-differential and fractional order differential equations considered in Chapters 2–6 with some random (stochastic) processes defined by stochastic differential equations. We assume that the reader is familiar with basic notions of probability theory and stochastic processes, such as a random variable, its density function, mathematical expectation, characteristic function, etc. Since we are interested only in applications of fractional order ΨDOSS, we do not discuss in detail facts on random processes that are already established and presented in other sources. For details of such notations and related facts we refer the reader to the book by Applebaum [App09] (or [IW81, Sat99]). We only mention some basic notations directly related to our discussions on fractional Fokker-Planck-Kolmogorov equations.

Random walk approximants of mixed and time-changed Lévy processes

Random walks are used to model various random processes in different fields. In this chapter we are only interested in random walks as approximating processes of some basic driving processes of stochastic differential equations discussed in the previous chapter. There is a vast literature (see, e.g., [GK54, Don52, Bil99, Taq75, GM98-1, GM01, MS01]) devoted to approximation of various basic stochastic processes like Brownian motion, fractional Brownian motion, Levy processes, and their time-changed counterparts. In the context of approximation, the question in what sense a random walk approximates (or converges to) an associated stochastic process becomes important. We will be interested only in the convergence in the sense of finite-dimensional distributions, which is equivalent to the locally uniform convergence of corresponding characteristic functions (see, e.g., [Bil99]).

Initial and boundary value problems for fractional order differential equations

In this chapter we will discuss boundary value problems for fractional order differential and pseudo-differential equations. For methodological clarity we first consider in detail the Cauchy problem for pseudo-differential equations of time-fractional order β, \(m - 1 <\beta 0,\ x \in \mathbb{R}^{n},& &{}\end{array}