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.