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Mark M. Meerschaert

Publications and source records attributed to Mark M. Meerschaert.

At least 19 recordsLinked to original sources
arXiv2019

What Is the Fractional Laplacian?

The fractional Laplacian in R^d has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function u(x) must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. In contrast, the spectral definition requires only the standard local boundary condition. These differences, among others, lead us to ask the question: "What is the fractional Laplacian?" We compare several commonly used definitions of the fractional Laplacian (the Riesz, spectral, directional, and horizon-based nonlocal definitions), and we use a joint theoretical and computational approach to examining their different characteristics by studying solutions of related fractional Poisson equations formulated on bounded domains. In this work, we provide new numerical methods as well as a self-contained discussion of state-of-the-art methods for discretizing the fractional Laplacian, and we present new results on the differences in features, regularity, and boundary behaviors of solutions to equations posed with these different definitions. We present stochastic interpretations and demonstrate the equivalence between some recent formulations. Through our efforts, we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications.

math.NA↗
arXiv2019

Space-Time Duality and High-Order Fractional Diffusion

Super-diffusion, characterized by a spreading rate $t^{1/α}$ of the probability density function $p(x,t) = t^{-1/α} p \left( t^{-1/α} x , 1 \right)$, where $t$ is time, may be modeled by space-fractional diffusion equations with order $1 < α< 2$. Some applications in biophysics (calcium spark diffusion), image processing, and computational fluid dynamics utilize integer-order and fractional-order exponents beyond than this range ($α> 2$), known as high-order diffusion, or hyperdiffusion. Recently, space-time duality, motivated by Zolotarev's duality law for stable densities, established a link between time-fractional and space-fractional diffusion for $1 < α\leq 2$. This paper extends space-time duality to fractional exponents $1<α\leq 3$, and several applications are presented. In particular, it will be shown that space-fractional diffusion equations with order $2<α\leq 3$ model sub-diffusion and have a stochastic interpretation. A space-time duality for tempered fractional equations, which models transient anomalous diffusion, is also developed.

cond-mat.stat-mech↗
arXiv2018

The fractional d'Alembert's formulas

In this paper we develop generalized d'Alembert's formulas for abstract fractional integro-differential equations and fractional differential equations on Banach spaces. Some examples are given to illustrate our abstract results, and the probability interpretation of these fractional d'Alembert's formulas are also given. Moreover, we also provide d'Alembert's formulas for abstract fractional telegraph equations.

math.FA↗
arXiv2018

Relaxation patterns and semi-Markov dynamics

Exponential relaxation to equilibrium is a typical property of physical systems, but inhomogeneities are known to distort the exponential relaxation curve, leading to a wide variety of relaxation patterns. Power law relaxation is related to fractional derivatives in the time variable. More general relaxation patterns are considered here, and the corresponding semi-Markov processes are studied. Our method, based on Bernstein functions, unifies three different approaches in the literature.

math.PR↗
arXiv2018

Semi-fractional diffusion equations

It is well known that certain fractional diffusion equations can be solved by the densities of stable Lévy motions. In this paper we use the classical semigroup approach for Lévy processes to define semi-fractional derivatives, which allows us to generalize this statement to semistable Lévy processes. A Fourier series approach for the periodic part of the corresponding Lévy exponents enables us to represent semi-fractional derivatives by a Grünwald-Letnikov type formula. We use this formula to calculate semi-fractional derivatives and solutions to semi-fractional diffusion equations numerically. In particular, by means of the Grünwald-Letnikov type formula we provide a numerical algorithm to compute semistable densities.

math.PR↗
arXiv2017

A Unified Spectral Method for FPDEs with Two-sided Derivatives; Stability, and Error Analysis

We present the stability and error analysis of the unified Petrov-Galerkin spectral method, developed in \cite{samiee2017Unified}, for linear fractional partial differential equations with two-sided derivatives and constant coefficients in any ($1+d$)-dimensional space-time hypercube, $d = 1, 2, 3, \cdots$, subject to homogeneous Dirichlet initial/boundary conditions. Specifically, we prove the existence and uniqueness of the weak form and perform the corresponding stability and error analysis of the proposed method. Finally, we perform several numerical simulations to compare the theoretical and computational rates of convergence.

cs.CE↗
arXiv2017

Boundary Conditions for Fractional Diffusion

This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.

math.AP↗
arXiv2016

Reflected Spectrally Negative Stable Processes and their Governing Equations

This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fractional reflecting boundary condition at the origin. Then we apply numerical methods to explicitly compute the transition density of this space-inhomogeneous Markov process, for any starting point, to any desired degree of accuracy. Finally, we discuss an application to fractional Cauchy problems, which involve a positive Caputo fractional derivative in time.

math.PR↗
arXiv2016

Domain and range symmetries of operator fractional Brownian fields

An operator fractional Brownian field (OFBF) is a Gaussian, stationary increment R^n-valued random field on R^m that satisfies the operator self-similarity property {X(c^E t)}_{t in R^m} L= {c^H X(t)}_{t in R^m}, c > 0, for two matrix exponents (E,H). In this paper, we characterize the domain and range symmetries of OFBF, respectively, as maximal groups with respect to equivalence classes generated by orbits and, based on a new anisotropic polar-harmonizable representation of OFBF, as intersections of centralizers. We also describe the sets of possible pairs of domain and range symmetry groups in dimensions (m,1) and (2,2).

math.PR↗
arXiv2016

Exponents of operator self-similar random fields

If X(c^E t) and c^H X(t) have the same finite-dimensional distributions for some linear operators E and H, we say that the random vector field X(t) is operator self-similar. The exponents E and H are not unique in general, due to symmetry. This paper characterizes the possible set of range exponents H for a given domain exponent, and conversely, the set of domain exponents E for a given range exponent.

math.PR↗
arXiv2016

Asymptotic behavior of semistable Lévy exponents and applications to fractal path properties

This paper proves sharp bounds on the tails of the Lévy exponent of an operator semistable law on $\mathbb R^d$. These bounds are then applied to explicitly compute the Hausdorff and packing dimensions of the range, graph, and other random sets describing the sample paths of the corresponding operator semi-selfsimilar Lévy processes. The proofs are elementary, using only the properties of the Lévy exponent, and certain index formulae.

math.PR↗
arXiv2016

Space-time fractional Dirichlet problems

This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes time-changed by an inverse stable subordinator whose index equals the order of the fractional time derivative. Some applications are given, to demonstrate how to specify a well-posed Dirichlet problem for space-time fractional diffusions in one or several variables. This solves an open problem in numerical analysis.

math.PR↗
arXiv2014

Extreme value statistics for truncated Pareto-type distributions

Recently attention has been drawn to practical problems with the use of unbounded Pareto distributions, for instance when there are natural upper bounds that truncate the probability tail. Aban, Meerschaert and Panorska (2006) derived the maximum likelihood estimator for the Pareto tail index of a truncated Pareto distribution with a right truncation point $T$. The Hill (1975) estimator is then obtained by letting $T \to \infty$. The problem of extreme value estimation under right truncation was also introduced in Nuyts (2010) who proposed a similar estimator for the tail index and considered trimming of the number of extreme order statistics. Given that in practice one does not always know whether the distribution is truncated or not, we discuss estimators for the Pareto index and extreme quantiles both under truncated and non-truncated Pareto-type distributions. We also propose a truncated Pareto QQ-plot in order to help deciding between a truncated and a non-truncated case. In this way we extend the classical extreme value methodology adding the truncated Pareto-type model with truncation point $T \to \infty$ as the sample size $n \to \infty$. Finally we present some practical examples, asymptotics and simulation results.

math.ST↗
arXiv2014

Semi-Markov approach to continuous time random walk limit processes

Continuous time random walks (CTRWs) are versatile models for anomalous diffusion processes that have found widespread application in the quantitative sciences. Their scaling limits are typically non-Markovian, and the computation of their finite-dimensional distributions is an important open problem. This paper develops a general semi-Markov theory for CTRW limit processes in $\mathbb{R}^d$ with infinitely many particle jumps (renewals) in finite time intervals. The particle jumps and waiting times can be coupled and vary with space and time. By augmenting the state space to include the scaling limits of renewal times, a CTRW limit process can be embedded in a Markov process. Explicit analytic expressions for the transition kernels of these Markov processes are then derived, which allow the computation of all finite dimensional distributions for CTRW limits. Two examples illustrate the proposed method.

math.PR↗
arXiv2011

Space-time fractional diffusion on bounded domains

Fractional diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues. They are used in physics to model anomalous diffusion. This paper develops strong solutions of space-time fractional diffusion equations on bounded domains, as well as probabilistic representations of these solutions, which are useful for particle tracking codes.

math.PR↗
arXiv2011

Transient anomalous sub-diffusion on bounded domains

This paper develops strong solutions and stochastic solutions for the tempered fractional diffusion equation on bounded domains. First the eigenvalue problem for tempered fractional derivatives is solved. Then a separation of variables, and eigenfunction expansions in time and space, are used to write strong solutions. Finally, stochastic solutions are written in terms of an inverse subordinator.

math.PR↗
arXiv2011

The fractional Poisson process and the inverse stable subordinator

The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also establishes an interesting connection between the fractional Poisson process and Brownian time.

math.PR↗