Erd\'elyi-Kober Fractional Diffusion

The aim of this Short Note is to highlight that the {\it generalized grey Brownian motion} (ggBm) is an anomalous diffusion process driven by a fractional integral equation in the sense of Erd\'elyi-Kober, and for this reason here it is proposed to call such family of diffusive processes as {\it Erd\'elyi-Kober fractional diffusion}. The ggBm is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion. This class is made up of self-similar processes with stationary increments and it depends on two real parameters: $0 < \alpha \le 2$ and $0 < \beta \le 1$. It includes the fractional Brownian motion when $0 < \alpha \le 2$ and $\beta=1$, the time-fractional diffusion stochastic processes when $0 < \alpha=\beta <1$, and the standard Brownian motion when $\alpha=\beta=1$. In the ggBm framework, the Mainardi function emerges as a natural generalization of the Gaussian distribution recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.Comment: Accepted for publication in Fractional Calculus and Applied Analysi

Short note on the emergence of fractional kinetics

In the present Short Note an idea is proposed to explain the emergence and the observation of processes in complex media that are driven by fractional non-Markovian master equations. Particle trajectories are assumed to be solely Markovian and described by the Continuous Time Random Walk model. But, as a consequence of the complexity of the medium, each trajectory is supposed to scale in time according to a particular random timescale. The link from this framework to microscopic dynamics is discussed and the distribution of timescales is computed. In particular, when a stationary distribution is considered, the timescale distribution is uniquely determined as a function related to the fundamental solution of the space-time fractional diffusion equation. In contrast, when the non-stationary case is considered, the timescale distribution is no longer unique. Two distributions are here computed: one related to the M-Wright/Mainardi function, which is Green's function of the time-fractional diffusion equation, and another related to the Mittag-Leffler function, which is the solution of the fractional-relaxation equation

The M-Wright function in time-fractional diffusion processes: a tutorial survey

In the present review we survey the properties of a transcendental function of the Wright type, nowadays known as M-Wright function, entering as a probability density in a relevant class of self-similar stochastic processes that we generally refer to as time-fractional diffusion processes. Indeed, the master equations governing these processes generalize the standard diffusion equation by means of time-integral operators interpreted as derivatives of fractional order. When these generalized diffusion processes are properly characterized with stationary increments, the M-Wright function is shown to play the same key role as the Gaussian density in the standard and fractional Brownian motions. Furthermore, these processes provide stochastic models suitable for describing phenomena of anomalous diffusion of both slow and fast type.Comment: 32 pages, 3 figure

Nonlinear Time-Fractional Differential Equations in Combustion Science

MSC 2010: 34A08 (main), 34G20, 80A25The application of Fractional Calculus in combustion science to model the evolution in time of the radius of an isolated premixed flame ball is highlighted. Literature equations for premixed flame ball radius are rederived by a new method that strongly simplifies previous ones. These equations are nonlinear time-fractional differential equations of order 1/2 with a Gaussian underlying diffusion process. Extending the analysis to self-similar anomalous diffusion processes with similarity parameter ν/2 > 0, the evolution equations emerge to be nonlinear time-fractional differential equations of order 1−ν/2 with a non-Gaussian underlying diffusion process

Development of an Ex Vivo Organ Culture Technique to Evaluate Probiotic Utilization in IBD

The consistent technical and conceptual progress in the study of the microbiota has led novel impulse to the research for therapeutical application of probiotic bacteria in human pathologies, such as inflammatory bowel disease (IBD). Considering the heterogenous results of probiotics in clinical studies, the model of translational medicine may lead to a more specific and efficacious utilization of probiotic bacteria in IBD. In this regard, the selection and utilization of appropriate experimental models may drive the transition from pure in vitro systems to practical clinical application. We developed a simple and reproducible ex vivo organ culture method with potential utilization for the evaluation of probiotic bacteria efficacy in IBD patients

Distance, bank heterogeneity and entry in local banking markets

We examine the determinants of entry into Italian local banking markets during the period 1991-2002 and build a simple model in which the probability of branching in a new market depends on the features of both the local market and the potential entrant. Our econometric findings show that, all else being equal, banks are more likely to expand into those markets that are closest to their pre-entry locations. We also find that large banks are more able to cope with distance-related entry costs than small banks. Finally, we show that banks have become increasingly able to open branches in distant markets, probably due to the advent of information and communication technologies.entry, barriers to entry, local banking markets, geographical distance.