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Zolotarev proved a duality result that relates stable densities with different indices. In this paper, we show how Zolotarev duality leads to some interesting results on fractional diffusion. Fractional diffusion equations employ fractional derivatives in place of the usual integer order derivatives. They govern scaling limits of random walk models, with power law jumps leading to fractional derivatives in space, and power law waiting times between the jumps leading to fractional derivatives in time. The limit process is a stable Levy motion that models the jumps, subordinated to an inverse stable process that models the waiting times. Using duality, we relate the density of a spectrally negative stable process with index $1<alpha<2$ to the density of the hitting time of a stable subordinator with index $1/alpha$, and thereby unify some recent results in the literature. These results also provide a concrete interpretation of Zolotarev duality in terms of the fractional diffusion model.
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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.
Super-diffusion, characterized by a spreading rate $t^{1/alpha}$ of the probability density function $p(x,t) = t^{-1/alpha} p left( t^{-1/alpha} x , 1 right)$, where $t$ is time, may be modeled by space-fractional diffusion equations with order $1 < alpha < 2$. Some applications in biophysics (calcium spark diffusion), image processing, and computational fluid dynamics utilize integer-order and fractional-order exponents beyond than this range ($alpha > 2$), known as high-order diffusion, or hyperdiffusion. Recently, space-time duality, motivated by Zolotarevs duality law for stable densities, established a link between time-fractional and space-fractional diffusion for $1 < alpha leq 2$. This paper extends space-time duality to fractional exponents $1<alpha leq 3$, and several applications are presented. In particular, it will be shown that space-fractional diffusion equations with order $2<alpha leq 3$ model sub-diffusion and have a stochastic interpretation. A space-time duality for tempered fractional equations, which models transient anomalous diffusion, is also developed.
We study the inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion with Hurst index $Hin(0,1)$. With the aid of a novel estimate, by using the operator approach we propose regularity analyses for the direct problem. Then we provide a reconstruction scheme for the source terms $f$ and $g$ up to the sign. Next, combining the properties of Mittag-Leffler function, the complete uniqueness and instability analyses are provided. Its worth mentioning that all the analyses are unified for $Hin(0,1)$.
A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. The fundamental solution (for the {Cauchy} problem) of the fractional diffusion equations can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to a given fractional diffusion equation.
We develop the theory of strong stationary duality for diffusion processes on compact intervals. We analytically derive the generator and boundary behavior of the dual process and recover a central tenet of the classical Markov chain theory in the diffusion setting by linking the separation distance in the primal diffusion to the absorption time in the dual diffusion. We also exhibit our strong stationary dual as the natural limiting process of the strong stationary dual sequence of a well chosen sequence of approximating birth-and-death Markov chains, allowing for simultaneous numerical simulations of our primal and dual diffusion processes. Lastly, we show how our new definition of diffusion duality allows the spectral theory of cutoff phenomena to extend naturally from birth-and-death Markov chains to the present diffusion context.
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