A continuous time random walk (CTRW) is a random walk subordinated to a renewal process, used in physics to model anomalous diffusion. Transition densities of CTRW scaling limits solve fractional diffusion equations. This paper develops more general limit theorems, based on triangular arrays,...

A scalar valued random field is called operator-scaling if for some dxd matrix E with positive real parts of the eigenvalues and some H0 we have where denotes equality of all finite-dimensional marginal distributions. We present a moving average and a harmonizable representation of stable...

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...

A stochastic process on a finite-dimensional real vector space is operator-self-similar if a linear time change produces a new process whose distributions scale back to those of the original process, where we allow scaling by a family of affine linear operators. We prove a spectral decomposition...

In this paper, we define and study a new class of random fields called harmonizable multi-operator scaling stable random fields. These fields satisfy a local asymptotic operator scaling property which generalizes both the local asymptotic self-similarity property and the operator scaling...