Fokker--Planck and Kolmogorov Backward Equations for Continuous Time Random Walk scaling limits

It is proved that the distributions of scaling limits of Continuous Time Random Walks (CTRWs) solve integro-differential equations akin to Fokker-Planck Equations for diffusion processes. In contrast to previous such results, it is not assumed that the underlying process has absolutely continuous laws. Moreover, governing equations in the backward variables are derived. Three examples of anomalous diffusion processes illustrate the theory.Comment: in Proceedings of the American Mathematical Society, Published electronically July 12, 201

A Vector-Valued Operational Calculus and Abstract Cauchy Problems.

Initial and boundary value problems for linear differential and integro-differential equations are at the heart of mathematical analysis. About 100 years ago, Oliver Heaviside promoted a set of formal, algebraic rules which allow a complete analysis of a large class of such problems. Although Heaviside\u27s operational calculus was entirely heuristic in nature, it almost always led to correct results. This encouraged many mathematicians to search for a solid mathematical foundation for Heaviside\u27s method, resulting in two competing mathematical theories: (a) Laplace transform theory for functions, distributions and other generalized functions, (b) J. Mikusinski\u27s field of convolution quotients of continuous functions. In this dissertation we will investigate a unifying approach to Heaviside\u27s operational calculus which allows us to extend the method to vector-valued functions. The main components are (a) a new approach to generalized functions, considering them not primarily as functionals on a space of test functions or as convolution quotients in Mikusinski\u27s quotient field, but as limits of continuous functions in appropriate norms, and (b) an asymptotic extension of the classical Laplace transform allowing the transform of functions and generalized functions of arbitrary growth at infinity. The mathematics are based on a careful analysis of the convolution transform $f \to k \star f.$ This is done via a new inversion formula for the Laplace transform, which enables us to extend Titchmarsh\u27s injectivity theorem and Foias\u27 dense range theorem for the convolution transform to Banach space valued functions. The abstract results are applied to abstract Cauchy problems. We indicate the manner in which the operational methods can be employed to obtain existence and uniqueness results for initial value problems for differential equations in Banach spaces

Brownian subordinators and fractional Cauchy problems

A Brownian time process is a Markov process subordinated to the absolute value of an independent one-dimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of processes, emerging as the scaling limits of continuous time random walks, involve subordination to the inverse or hitting time process of a classical stable subordinator. The resulting densities solve fractional Cauchy problems, an extension that involves fractional derivatives in time. In this paper, we will show a close and unexpected connection between these two classes of processes, and consequently, an equivalence between these two families of partial differential equations.Comment: 18 pages, minor spelling correction

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

Inhomogeneous Fractional Diffusion Equations

2000 Mathematics Subject Classification: Primary 26A33; Secondary 35S10, 86A05Fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and time. They are useful to model anomalous diffusion, where a plume of particles spreads in a different manner than the classical diffusion equation predicts. An initial value problem involving a space-fractional diffusion equation is an abstract Cauchy problem, whose analytic solution can be written in terms of the semigroup whose generator gives the space-fractional derivative operator. The corresponding time-fractional initial value problem is called a fractional Cauchy problem. Recently, it was shown that the solution of a fractional Cauchy problem can be expressed as an integral transform of the solution to the corresponding Cauchy problem. In this paper, we extend that results to inhomogeneous fractional diffusion equations, in which a forcing function is included to model sources and sinks. Existence and uniqueness is established by considering an equivalent (non-local) integral equation. Finally, we illustrate the practical application of these results with an example from groundwater hydrology, to show the effect of the fractional time derivative on plume evolution, and the proper specification of a forcing function in a time-fractional evolution equation.1. Partially supported by the Marsden Foundation in New Zealand 2. Partially supported by ACES Postdoctoral Fellowship, Nevada NSF EPSCoR RING TRUE II grant 3. Partially supported by NSF grants DMS-0139927 and DMS-0417869, and the Marsden Foundatio