Centre-of-Mass Like Superposition of Ornstein–Uhlenbeck Processes: A Pathway to Non-Autonomous Stochastic Differential Equations and to Fractional Diffusion

D’Ovidio, Mirko; Vitali, Silvia; Sposini, Vittoria; Sliusarenko, Oleksii; Paradisi, Paolo; Castellani, Gastone; Pagnini, Gianni

doi:10.1515/fca-2018-0074

Cited by 17 publications

(12 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A new direction in theoretical modelling of diffusion in complex media interpretes the anomalous character of the diffusion as a consequence of a very heterogeneous enviroment [7,8,13,21,43,44]. One type of such models is based on randomly scaled Gaussian processes (RSGP) and is sometimes refered to as Generalized Grey Brownian Motion (GGBM), or GGBM-like models.…”

Section: Introductionmentioning

confidence: 99%

Stochastic solutions of generalized time-fractional evolution equations

Bender,

Butko

2021

Preprint

View full text Add to dashboard Cite

We consider a general class of integro-differential evolution equations which includes the governing equation of the generalized grey Brownian motion and the time-and space-fractional heat equation. We present a general relation between the parameters of the equation and the distribution of the underlying stochastic processes, as well as discuss different classes of processes providing stochastic solutions of these equations. For a subclass of evolution equations, containing Saigo-Maeda generalized time-fractional operators, we determine the parameters of the corresponding processes explicitly. Moreover, we explain how self-similar stochastic solutions with stationary increments can be obtained via linear fractional Lévy motion for suitable pseudo-differential operators in space.

show abstract

Section: Introductionmentioning

confidence: 99%

Stochastic solutions of generalized time-fractional evolution equations

Bender,

Butko

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In [8], the results of of [49] and [20] for fBm were extended to the non-Gaussian case of ggBm by representing it via generalized grey Ornstein-Uhlenbeck processes, using that ggBm can be written as a product of a positive and time-independent random variable and a fBm [39]. A similar representation can be found in [14]. While recent progress is visible for the one-dimensional generalized grey Brownian motion, many results can not be carried over directly to a multi-dimensional case.…”

Section: Introductionmentioning

confidence: 99%

Stochastic analysis for vector-valued generalized grey Brownian motion

Böck¹,

Grothaus²,

Karlo³

2021

Preprint

View full text Add to dashboard Cite

In this article, we show that the standard vector-valued generalization of a generalized grey Brownian motion (ggBm) has independent components if and only if it is a fractional Brownian motion. In order to extend ggBm with independent components, we introduce a vector-valued generalized grey Brownian motion (vggBm). The characteristic function of the corresponding measure is introduced as the product of the characteristic functions of the one-dimensional case. We show that for this measure, the Appell system and a calculus of generalized functions or distributions are accessible. We characterize these distributions with suitable transformations and give a d-dimensional Donsker's delta function as an example for such distributions. From there, we show the existence of local times and self-intersection local times of vggBm as distributions under some constraints, and compute their corresponding generalized expectations. At the end, we solve a system of linear SDEs driven by a vggBm noise in d dimensions.

show abstract

“…Other authors follow a somewhat different approach based on random-scaled Gaussian processes (RSGPs) [38,[56][57][58][59], which are physically based on a recently proposed model where interparticle heterogeneity is explicity described through a population of scales characterizing the dynamical parameters of particle diffusive motion. This modeling approach has been denoted as heterogeneous ensemble of Brownian particles (HEBP) and has been developed on the basis of a Langevin model [57][58][59]. The HEBP model is then based on the Gaussian-Wiener process and, thus, on trajectories that are strongly continuous in the stochastic sense [60], while anomalous diffusion emerge as a consequence of heterogeneity.…”

Section: Introductionmentioning

confidence: 99%

Gaussian Processes in Complex Media: New Vistas on Anomalous Diffusion

et al. 2019

Self Cite

View full text Add to dashboard Cite

Normal or Brownian diffusion is historically identified by the linear growth in time of the variance and by a Gaussian shape of the displacement distribution. Processes departing from the at least one of the above conditions defines anomalous diffusion, thus a nonlinear growth in time of the variance and/or a non-Gaussian displacement distribution. Motivated by the idea that anomalous diffusion emerges from standard diffusion when it occurs in a complex medium, we discuss a number of anomalous diffusion models for strongly heterogeneous systems. These models are based on Gaussian processes and characterized by a population of scales, population that takes into account the medium heterogeneity. In particular, we discuss diffusion processes whose probability density function solves space-and time-fractional diffusion equations through a proper population of timescales or a proper population of length-scales. The considered modeling approaches are: the continuous time random walk, the generalized gray Brownian motion, and the time-subordinated process. The results show that the same fractional diffusion follows from different populations when different Gaussian processes are considered. The different populations have the common feature of a large spreading in the scale values, related to power-law decay in the distribution of population itself. This suggests the key role of medium properties, embodied in the population of scales, in the determination of the proper stochastic process underlying the given heterogeneous medium.

show abstract

Centre-of-Mass Like Superposition of Ornstein–Uhlenbeck Processes: A Pathway to Non-Autonomous Stochastic Differential Equations and to Fractional Diffusion

Cited by 17 publications

References 30 publications

Stochastic solutions of generalized time-fractional evolution equations

Stochastic solutions of generalized time-fractional evolution equations

Stochastic analysis for vector-valued generalized grey Brownian motion

Gaussian Processes in Complex Media: New Vistas on Anomalous Diffusion

Contact Info

Product

Resources

About