Invited speakers
Larbi Alili
We prove that a space inversion property of a standard Markov process X implies the existence of a Kelvin transform of X-harmonic, excessive and operator--harmonic functions and that the inversion property is inherited by Doob h-transforms. We determine new classes of processes having space inversion properties amongst those satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. Some examples are treated in details.
Mirko D'Ovidio
Fractional and anomalous diffusions have a long history. The terms fractional and anomalous have been considered with different meaning and in different contexts. By fractional diffusion we mean a diffusion in a medium with fractional dimension (fractals, for instance) whereas, by anomalous diffusions, according to the most significant literature, we refer to a motion whose mean squared displacement is proportional to a power of time. In this context we usually have the characterization given in terms of subdiffusion/superdiffusion or normal diffusion. Our aim is to pay exclusive attention to the probabilistic models for anomalous dynamics (not necessarily anomalous diffusions). The involved processes are guided by fractional (in time/space) equations and are obtained through random time changes.
We introduce a precise definition of delayed and rushed processes and provide some examples which are, in some cases, counter-intuitive.
We consider time changes given by subordinators and their inverse processes. Our analysis shows that, quite surprisingly, inverse processes are not necessarily leading to delayed processes.
József Lőrinczi
I will consider a class of non-local equations, assuming that their solutions have a global maximum, and discuss estimates on their position. Such points are of special interest as they describe where hot-spots settle on the long run or, in a probabilistic description, the region around which paths typically concentrate. I will address, on the one hand, the interplay of competing parameters whose compromise the resulting location is, and the impact of geometric constraints, on the other.
References
[1] A. Biswas and J. Lőrinczi. Universal constraints on the location of extrema of eigenfunctions
of non-local Schrödinger operators, arXiv:1710.11596, 2017 (under review)
[2] A. Biswas and J. Lőrinczi. Maximum principles and Aleksandrov-Bakelman-Pucci type
estimates for non-local Schrödinger equations with exterior conditions, arXiv:1711.09267,
2017 (under review)
[3]
A. Biswas and J. Lőrinczi.
Maximum principles for time-fractional Cauchy problems with
spatially non-local components, arXiv:1801.02349, 2018 (under review)
[4]
A. Biswas and J. Lőrinczi.
Ambrosetti-Prodi type results for Dirichlet problems of the
fractional Laplacian, arXiv:1803.08540, 2018 (under review)
Carlo Manzo
Cellular signaling is regulated by biochemical interactions that are ultimately controlled by molecular diffusion. Recent advances in fluorescence microscopy have allowed the visualization of single molecules in living cells at unprecedented spatiotemporal resolution, revealing that the heterogeneity of the cellular environment produces exotic molecular motions that deviate from Brownian behavior [1]. These findings have stimulated new questions about the mechanisms generating these phenomena, as well as regarding their implications for cell biology.
In this context, we have studied a transmembrane receptor involved in the capture of pathogens, which motion exhibits anomalous diffusion with signatures of weak ergodicity breaking [2]. Through the study of receptor mutants, we have been able to correlate the receptors motion to its molecular structures, lateral organization and interactions, thus establishing a link between nonergodicity and biological function. In addition, we have quantitatively interpreted the receptor dynamics through a stochastic model of random motion with random diffusivity on scale-free media [3,4], and we are attempting to gain further insight into the molecular causes of this complex diffusion.
Our work highlights the role of heterogeneity in cell membranes and proposes a connection with function regulation. In addition, our models offer a theoretical framework to interpret anomalous transport in complex media, such as those found, e.g., in soft condensed matter, geology, and ecology.
References
[1]C. Manzo, and M. F. Garcia-Parajo, Rep. Prog. Phys. 78:124601 (2015).
[2]C. Manzo, et al., Phys. Rev. X 5:011021 (2015).
[3]P. Massignan, et al., Phys. Rev. Lett. 112:150603 (2014).
[4]C. Charalambous, et al., Phys. Rev. E 95:032403 (2017).
Łukasz Płociniczak
We consider a one-dimensional nonlinear diffusion equation with nonlocal time, i.e. the temporal derivative is calculated via the Riemann-Liouville operator. This problem set on a half-line models moisture percolation in certain porous media such as construction materials.
The efficient and convergent numerical method can be constructed as follows. We cast our equation into the self-similar form with the use of the Erdelyi-Kober operator. Then, by a suitable transformation the free-boundary problem is transformed into the initial-value one. Lastly, the resulting integro-differential equation can be written as a nonlinear Volterra equation. In this way, a nonlinear PDE with free boundary can be solved by finding a solution of an ordinary Volterra equation. This quickens the calculations tremendously.
We also prove that a certain family of numerical schemes is convergent to the solution of the aforementioned Volterra equation. The main difficulty lies in the non-Lipschitzian character of the governing nonlinearity. In that case we cannot use the classical theory and have to proceed in other ways. In the talk we will describe the details of the proof.
Luz Roncal
We prove Hardy-type inequalities for fractional powers of the sublaplacian in the Heisenberg group. In order to get these inequalities, we study the extension problem associated to the sublaplacian. Solutions of the extension problem are written down explicitly and used to establish a trace Hardy inequality that will lead to a Hardy inequality with sharp constants.
Several new results concerning the extension problem in the Heisenberg group are also attained, including characterisations of all solutions of the extension problem satisfiying \(L^p\) integrability, and the study of the higher order extension problem.
This is a joint work with S. Thangavelu (Indian Institute of Science of Bangalore, India).
Laura Sacerdote
The coherence between the input and the output of the single units is a problem, sometimes underestimated, in network modeling.
An example in this direction is given by Integrate and Fire models used to describe the membrane potential dynamics of a neuron in a network. The focus of these models concerns the description of the inter-times between events (the InterSpike Intervals, ISIs), i.e. of the output of the neuron. This type of models describes the membrane potential evolution through a suitable stochastic process and the output of the neuron corresponds to the First Passage Time of the considered stochastic process through a boundary. However, the input mechanism determining the membrane potential dynamics often disregards its origin as function of the output of other units.
The seminal idea for these models goes back to 1964 when Gernstein and Mandelbrot proposed the Integrate and Fire model to account for the observed stable behavior of the Inter-spike Interval distribution. They suggested to model the membrane potential dynamics through a Wiener process in order to get the Inverse Gaussian distribution for the inter-times between the successive spikes of the neuron, i.e. its output. The use of the Wiener process was first motivated by its property to be the continuous limit of a random walk, later the randomized random walk was proposed to account for the continuous time characterizing the membrane potential dynamics. In this model, the arrival of inputs from the network determines jumps of fixed size on the membrane potential value. When the membrane potential attains a threshold value \(S\), the neuron releases a spike and the process restart anew. Furthermore, the inter-times between jumps are exponentially distributed.
Unfortunately, this last hypothesis is contradictory with the heavy tail distribution of the output, since the incoming inputs are output of other neurons. Later many variants of the original model appeared in the literature. Their aim was to improve the realism of the model but unfortunately they forgot the initial clue for it, the heavy tails of the observed output distribution.
However, the ISIs models (or their variants) are generally recognized to be a good compromise between the realism and its easy use and have been proposed to model large networks. These facts motivate us to rethink the ISIs model allowing heavy tail distributions both for the ISIs of the neurons surrounding the modeled neuron and for its output.
Here, we propose to start the model formulation from the main property, i.e. the heavy tails exhibited by the ISIs. This approach allows us to propose here an Integrate and Fire model coherent with these features. The ideal framework for this rethinking involves regularly varying random variables and vectors. We assume that each input to a unit corresponds to the output of one of \(N < \infty \) neurons of the network. The inter-times between spikes of the same neuron are independent random variables, with regularly varying distribution. Different neurons of the network are not independent, due to the network connections. The only hypothesis that we introduce to account for the dependence is very general, i.e. their ISIs determine a regularly varying vector. Based on these hypothesis we prove that the output inter-times of the considered neuron, described through an Integrate and Fire model, is a regularly varying random variable.
The next step of this modeling procedure requests a suitable rescaling of the obtained process to obtain a time fractional limit for the process describing the membrane potential evolution. We already have some result in this direction, allowing to write down the Laplace transform of the first passage time of the rescaled process through the threshold \(S\) but some mathematical step should be improved to account for the dependence between jumps times also for the limiting process.
References
[1] Bingham, N. H., Goldie, C. M. and Teugels, J. L., Regular variation, volume 27 of Encyclopedia of Mathematics
and its Applications. Cambridge University Press, Cambridge, 1989. ISBN 0-521-37943-1.
[2] Gal A., and Marom S. Entrainment of the Intrinsic Dynamics of Single Isola Neurons by Natural-Like Input,. The
Journal of Neurosciences 33 (18) pp. 7912-7918, 2013.
[3] Gernstein G.L., Mandelbrot B. Random walk models for the activity of a single neuron., Biophys. J. 4 pp. 41-68,
1964.
[4] Holden, A.V. A Note on Convolution and Stable Distributions in the Nervous System., Biol. Cybern. 20 pp.
171-173, 1975.
[5] Jessen, A. H., and Mikosch, T. Regularly varying functions. Publ. Inst. Math. (Beograd) (N.S.), 80(94):171–192,
2006. ISSN 0350-1302.
[6] Lindner B. Superposition of many independent spike trains is generally not a Poisson process, . Physical review E
73 pp. 022901, 2006.
[7] Kyprianou, A. Fluctuations of Lévy Processes with Applications., Springer-Verlag, 2014.
[8] Persi E., Horn D., Volman V., Segev R. and Ben-Jacob E. Modeling of Synchronized Bursting Events: the
importance of Inhomogeneity., Neural Computation 16 pp. 2577-2595, 2004.
[9] Tsubo Y., Isomura Y. and Fukai T. Power-Law Inter-Spike Interval Distributions infer a Conditional Maximization
of Entropy in Cortical Neurons,. PLOS Computational Biology 8,4 pp. e1002461, 2012.
Trifce Sandev
Recent developments in modeling of different physical and biological systems undoubtedly confirmed that fractional calculus is not just an exotic mathematical theory, as it might appeared at the dawn of its emergence. The present work aims to demonstrate this through physical examples of diffusion and quantum motion in two-dimensional comb-like structures, leading to fractional calculus description of the corresponding processes. Comb geometry is one of the most simple paradigm of a two-dimensional structure, where anomalous diffusion can be realized. The comb model, for the first time, has been applied to investigate anomalous diffusion in low-dimensional percolation clusters, and nowadays it has many applications in description of transport properties in porous dielectrics and low dimensional composites, turbulent diffusion on a comb, tracer dynamics in subsurface aquifers, and anomalous diffusion in spiny dendrites, to name but a few. In this presentation we will show how the time fractional diffusion and Schrödinger equations can be derived by projection of the two dimensional comb dynamics in the one-dimensional configuration space. The anomalous character of the corresponding process can be described by the fact that the particle which moves along the main backbone channel can get trapped in the fingers of the comb. We will also show how the anomalous diffusion along the main backbone depends on the fractal structure of the comb, resulting in natural appearance of fractional derivatives in the corresponding equations for the probability density function. The connection between the anomalous diffusion exponent and the fractal dimension of the comb/mesh structures has been recently experimentally observed in the anomalous transport through porous structurally inhomogeneous media.
Enrico Scalas
Using two simple examples, the continuous-time random walk as well as a two state Markov chain, the relation between generalized anomalous relaxation equations and semi-Markov processes is illustrated. This relation is then used to discuss continuous-time random statistics in a general setting, for statistics of convolution-type. Two examples are presented in some detail: the sum statistic and the maximum statistic.
Alessandro Taloni
The Generalized Elastic Model accounts for the dynamics of several physical systems, such as polymers, fluctuating interfaces, growing surfaces, membranes, proteins and file systems among others. We derive the fractional stochastic equation governing the motion of a probe particle (tracer) in such kind of systems. This Langevin equation involves the use of fractional derivative in time and satisfies the Fluctuation-Dissipation relation, it goes under the name of Fractional Langevin Equation. Within this framework the spatial correlations appearing in the Generalized Elastic Model are translated into time correlations described by the fractional derivative together with the space-time correlations of the fractional Gaussian noise. We derive the exact scaling analytical form of several physical observables such as structure factors, roughness and mean square displacement. Special attention will be devoted to the dependence on initial conditions and linear-response relations in the case of an applied potential.
Enrico Valdinoci
We consider a nonlocal equation driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way.
In particular, heteroclinic, homoclinic and chaotic trajectories are constructed.
This result regarding symbolic dynamics in a fractional framework is part of a study of the Peierls-Nabarro model for crystal dislocations. The associated evolution equation can be studied in the mesoscopic and macroscopic limit. Namely, the dislocation function has the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium.
These dislocation points evolve according to the external stress and an interior potential, which can be either repulsive or attractive, depending on the relative orientations of the dislocations. For opposite orientations, collisions occur, after which the system relaxes exponentially fast.
Zoran Vondraček
Let \(Z\) be an isotropic stable process in the Euclidean space. The process \(Z\) is killed upon exiting an open set \(D\) and the killed process is then subordinated by an independent \(\gamma\)-stable subordinator, \(0<\gamma <1\). The resulting process is a Hunt process in \(D\). In this talk, I will discuss several potential theoretical properties of this process such as Harnack inequality for nonnegative harmonic functions, the Carleson estimate, Green function and jumping kernel estimates in smooth sets \(D\), and in particular, the boundary Harnack principle. Surprisingly, it turns out the BHP holds only if \(1/2<\gamma<1\). This is joint work with Panki Kim and Renming Song.
Speakers
Jorge Cayama
We develop a pseudo-spectral method to solve initial-value problems associated to PDEs involving the fractional Laplacian operator acting on the whole real line (see [1]). After a suitable representation of the operator, we perform the change of variable \(x = L\cot(s)\), \(L > 0\), to transform the real line \(\mathbb{R}\) into the interval \([0, \pi]\), where a Fourier expansion of the solution \(u(x(s))\) can be applied. We approximate the fractional Laplacian by means of the midpoint quadrature rule, improving the results with Richardson’s extrapolation, similarly as in [2]. This method deals accurately and efficiently with problems posed on \(\mathbb{R}\), and avoids truncating the domain (which requires introducing artificial
boundary conditions). In order to illustrate its applicability, we have simulated the evolution of the following non-local Fisher-KPP [3] and ZFK-Nagumo [4] models.
Keywords: Fractional Laplacian, Pseudo-spectral methods, Fourier transform, Chebyshev Polynomials.
References
[1] M. Kwaśnicki. Ten equivalent definitions of the fractional Laplace operator. Fractional Calculus and Applied Analysis, 20(1):7–51, 2017.
[2] F. de la Hoz and C. M. Cuesta. A pseudo-spectral method for a non-local KdV-Burgers equation posed on \(\mathbb{R}\). Journal of Computational Physics, 311:45–61, 2016.
[3] R. A. Fisher. The wave of advance of advantageous genes. Annals. of Eugenics, 7:355–369, 1937.
[4] Y. B. Zel’dovich and D. A. Frank-Kamenetsky. Towards the theory of uniformly propagating flames. Doklady AN SSSR, 19:693–697, 1938.
[5] J. P. Boyd. Chebyshev and Fourier Spectral Method. Springer–Verlag, XVI, 1989.
Serena Dipierro
We discuss some recent results on nonlocal phase transitions modelled by the fractional Allen-Cahn equation, also in connection with the surfaces minimising a nonlocal perimeter functional. In particular, we consider the "genuinely nonlocal regime" in which the difusion operator is of order less than 1 and present some rigidity and symmetry results.
Ifan Johnston
In 1967 Aronson showed that the fundamental solution of the heat equation for a second order uniformly elliptic operator in divergence form satisfies two-sided Gaussian estimates. Using this celebrated result, we investigate two-sided estimates for the fundamental solution of the fractional analogues of the heat equation, where one replaces the time derivative with a Caputo fractional derivative of order \(\beta \in (0, 1)\) and also replace the second order elliptic operator with a homogeneous pseudo-differential operator. The starting point for these estimates is given by a formula, which is due to Zolotarev, that links Mittag-Leffler functions with \(\beta\)-stable densities via the Laplace transform. Probabilistically speaking, the solution of such fractional evolution equations is typically some time-changed Brownian motion, or time-changed stable process. This is joint work with Vassili Kolokoltsov.
Petra Lazić
In this talk, I will discuss ergodic properties of diffusion processes focusing on the rate of convergence of the marginals of the process to the invariant measure with respect to the total variation distance and Wasserstein distance. In particular, I will present sharp conditions in terms of the coefficients of the process (generator) ensuring subexponential rate of convergence. I will also discuss ergodic properties of a class of jump processes obtained through subordination of diffusion processes.
Pedro J. Miana
In this talk, we present a complete spectral research of generalized Cesàro operators on Sobolev-Lebesgue sequence spaces. The main idea is to subordinate such operators to suitable \(C_0\) -semigroups on these sequence spaces. We introduce that family of sequence spaces using the fractional finite differences and we prove some structural properties similar to classical Lebesgue sequence spaces. In order to show the main results about fractional finite differences, we state equalities involving sums of quotients of Euler's Gamma functions. Finally, we display some graphical representations of the spectra of generalized Cesàro operators. Main results of this talk are included in a joint paper with L. Abadias ([1]).
References
[1] L. Abadias and P.J. Miana. Generalized Cesàro operators, fractional finite differences and Gamma functions. J. Funct. Anal., 274 , (2018), 1424--1465.
Partially supported by Project MTM2016-77710-P, DGI-FEDER, of the MCYTS and Project E-64, D.G. Aragón, Universidad de Zaragoza, Spain.
Federico Polito
Fractionality in continuous time is usually achieved by suitable time-changes and it is commonly seen as a tool to extend Markov processes to models in which the presence of persistent memory is taken into consideration. Interestingly enough, these models are non-Markov, still they represent a class of processes retaining a certain mathematical tractability. Even though the literature about continuous-time fractional processes is vast and growing, only few studies on their discrete-time counterparts have appeared so far. In this talk we present a theory for processes in discrete time admitting in some cases persistent memory. This is achieved by considering discrete infinite divisibility of random variables and defining time-changes resembling and actually converging to inverse subordinators. An example of a discrete-time renewal process having as a scaling limit the time-fractional Poisson process is described. Finally, as a possible application, a time-changed preferential attachment model of random graph is constructed and analyzed highlighting the differences with the classical model.
The talk collects joint works with Angelica Pachon and Costantino Ricciuti.
Acknowledgement
F. Polito has been partially supported by the projects Memory in Evolving Graphs (Compagnia di San Paolo/Universita di Torino) and Sviluppo e analisi di processi Markoviani e non Markoviani con applicazioni (Universit\`a di Torino).
Xabier Telleria-Allika
Arbitrary order calculus is a natural generalisation of usual calculus in which the order of differentiation and integration operators in not restricted to integer numbers. In engineering fractional order derivatives are used for describing the behaviour of materials with memory (i.e. viscoelastic materials) due to the fact that these materials lay somewhere in between Hookean springs and Newtonean fluids[1,2]; there are also many dynamical systems which can be better described when arbitrary order derivatives are included [3].
Lanskin [4] formulated the first Fractional Schrödinger Equation (FSE) along with the Fractional Continuation Equation in 2002; however, we are still far from fully understanding the effect pf the FSE on physical properties such as: Tunnelling [5], Diffraction [6] and Scattering [7]. Due to the properties of fractional derivatives, many jobs have been done in which relativistic properties and effects of extrinsic magnetic fields are obtained by incorporating an arbitrary order to the kinetic energy in the Hamiltonian [8,9].
Further studies of the FSE applied on astrophysically interesting systems such as \(H_2^+\) [10] and even hydrogen atom [11] seem to be promising. We shall take the FSE for a particle in a ring (1) as a first step into this world for which the eigenvalues are (2) and the eigenfunctions (3)
\(
\left[\frac{1}{2mr^2}\right]^{\alpha-1}\left[i\hslash \partial_\theta \right]^\alpha \Psi_\alpha(\theta;r)=\lambda_\alpha\Psi_\alpha(\theta;r) \tag{1}
\)
\(
\lambda_\alpha = \left[\frac{1}{2mr^2}\right]^{1-\alpha} N^\alpha \hslash^\alpha \exp \left[ i \pi \alpha \left(n+1\right)\right]\; | \; N, \; n \in \mathbb{N}+\{0\} \tag{2}
\)
\(
\Psi_\alpha(\theta;r)=C\exp \left(-i N^\alpha\left[ \frac{\hslash}{2mr^2}\right]^{\alpha-1}\exp \left(i \pi \alpha \left(n+1\right)\right)\theta \right). \tag{3}
\)
References
[1] M. Stiassnie, 1979, Appl. Math. Modelling, 3, 300.
[2] M. Du et al., 2013, Scientific Reports, 3, 3431.
[3] V. E. Tarasov, 2013, Int. J. Mod. Phys. B, 2013, 9, 1330005.
[4]N. Laskin, Physics Review E, 2000, 66, 056108.
[5] E. Capelas et al., 2011, J. Phys. A 44, 185303.
[6] Y. Zhang et al., 2015, Scientific Reports 6, 23645.
[7] A. Liemert, 2016, Mathematics, 4, 31.
[8] J. Lorinczi and J. Malecki, 2012, J. Diff. Eq., 253, 2846.
[9] J. Blackledge and B. Babajanov, 2013, Math. Aeterna, 3, 601.
[10] A. Turbiner et al., 1999, JETP Letters, 11, 69.
[11] A. I. Arbab, 2012, J. Modern Physics, 3, 1737.
Lorenzo Toniazzi
Consider the Caputo evolution equation (EE) \(\partial^\beta_t u = \Delta u\) with initial condition \(\phi\) on \(\{0\}\times\mathbb R^d\). As it is well known, the solution reads \(u(t, x) = \mathbf{E}_x\left[\phi (B_{E_t})\right]\). Here \(B_t\) is a Brownian motion and the independent time change \(E_t\) is an inverse \(\beta\)-stable subordinator. This is a popular model for subdiffusion [6], with remarkable universality properties [1]. We substitute the Caputo derivative \(\partial^\beta_t\) with the Marchaud derivative. This results in a natural extension of the Caputo EE featuring a time-nonlocal initial condition \(\phi\) on \((-\infty, 0] \times \mathbb{R}^d\). We derive the new stochastic representation for the solution, namely \(u(t, x) = \mathbf{E}_x\left[\phi(-W_t, B_{E_t})\right]\). This stochastic representation has a pleasing interpretation due to the non-obvious presence of \(W_t\). Here \(W_t\) denotes the waiting/trapping time of the subdiffusion \(B_{E_t}\). We discuss classical-wellposedness for the space-fractional case, following [7]. Additionally, following [3, 4], we discuss weak-wellposedness for the respective extensions of Caputo-type EEs (such as in [2, 5]).
References
[1] Barlow, Martin T.; Cerny, Jiri (2011). Probability theory and related fields, 149.3-4: 639-673.
[2] Chen Z-Q., Kim P., Kumagai T., Wang J. (2017). arXiv:1708.05863.
[3] Du Q., Yang V., Zhou Z. (2017). Discrete and
continuous dynamical systems series B, Vol 22, n. 2.
[4] Du Q., Toniazzi L., Zhou Z. (2018).
Preprint. Expected submission date: Sept. 2018.
[5] Hernández-Hernández, M.E., Kolokoltsov, V.N., Toniazzi, L. (2017). Chaos, Solitons & Fractals, 102, 184-196.
[6] Meerschaert, M.M., Sikorskii, A. (2012). Stochastic Models for Fractional Calculus, De Gruyter
Studies in Mathematics, Book 43.
[7] Toniazzi L. (2018). To appear in J Math Anal Appl. arXiv:1805.02464.
Silvia Vitali
Anomalous diffusion is widely observed in biological systems. Lot of efforts have been dedicated
to derive models in agreement with all the statistical features emerging from data [1, 2], but an exhaustive description is still missing.
We derive a stochastic diffusion model based on a Langevin approach, characterized by additive noise and linear friction force [3].
The complexity of the medium is parametrized by a population of the parameters, the relaxation time and diffusivity.
For proper distributions of these parameters, both Gaussian anomalous diffusion,
fractional diffusion and its generalizations can be retrieved, but characterized by a superdiffusive regime by model construction.
The inclusion of a confining potential, for example by considering the harmonic Langevin oscillator, permits to switch the process to a subdiffusive regime.
This anomalous diffusive behaviour can be reflected in the motion of the center of mass of an heterogeneous ensamble of particles [4]
and the motion of an inert tracer globally connected with such heterogeneous mesoscopic surrounding.
References
[1] M. Mura, G. Pagnini. 2008 Characterizations and simulations of a
class of stochastic processes to model anomalous diffusion. J. Phys.
A: Math. Theor. \textbf{41}, 285003.
[2] D. Molina-Garciá, T. M. Pham, P. Paradisi, C. Manzo, G. Pagnini.
2016. Fractional kinetics emerging from ergodicity breaking in random
media Phys. Rev. E \textbf{94}, 052147.
[3] S. Vitali, V. Sposini, O. Sliusarenko, P. Paradisi, G. Castellani, G.
Pagnini. 2018. J. R. Soc. Interface \textbf{15}: 20180282.
http://dx.doi.org/10.1098/rsif.2018.0282
[4] M. D’Ovidio, S. Vitali, V. Sposini, O. Sliusarenko, P. Paradisi,
G. Castellani, G. Pagnini. 2018. Centre-of-mass like superposition
of Ornstein-Uhlenbeck processes: a pathway to non-autonomous
stochastic differential equations and to fractional diffusion. Submit-
ted. https://arxiv.org/abs/1806.11351.