Program

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.

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.

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.

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.

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.

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.

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.

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.