In recent years, time-fractional Schrödinger equations received some attention for their ability to describe non-Markov quantum phenomena. Nevertheless, different models have to be proposed in order to reproduce some analytic feature fo the equations. Unlike the classical case, it has been shown that the Wick-rotation construction of a time-fractional Schrödinger equation does not coincide with the one arising from a pure path-integral apprach. Furthermore, in the free particle case, these models exhibit different anomalous features, that, in any case, challenge the usual interpretation of the Schrödinger equation. In this talk, we will consider a toy model, i.e. the harmonic oscillator, and we will show that these anomalous features prevale even in presence of a potential. Furthermore, thanks to the pure discrete spectrum of the considered operator, we are able to relate the anomalous phenomena to the improved stability region implied by fractional Caputo derivatives.

These results are part of an ongoing joint work with Nikolai Leonenk from Cardiff university.

This talk surveys a unified mathematical framework for competitive exclusion, stability, and Lyapunov theory in population dynamics and ecology, based on reaction-network and mathematical epidemiology methods. The central objects are minimal siphons, relay graphs, and admissible Lyapunov cones, which are used to analyse local and global dynamics across several general classes of nonlinear systems.

Part I surveys some essential definitions, and their application to generalized Lotka--Volterra systems.

Second part provides our recent results showing that the famous next generation matrix (NGM) theorem is a consequence of the triangular block structure of the Jacobian on any siphon face, and that furthermore, the transversal Jacobian block is always Metzler. Part three exemplifies the theory through a two-rumor online social network model.

Grey measures provide a flexible framework for constructing non-Gaussian stochastic models while retaining a tractable analytic structure. This talk presents different developments of this approach, ranging from infinite-dimensional analysis to isotropic random fields on the sphere.

The first one concerns the Gamma-grey noise, a non-Gaussian measure on an infinite-dimensional space; this construction leads to a class of Gamma-grey stochastic processes, including fractional and tempered models for anomalous diffusion.

We also introduce Grey spherical random fields, a new class of non-Gaussian isotropic fields constructed through Mittag–Leffler mixing and multivariate subordinators. Their spherical harmonic coefficients admit a Gaussian variance-mixture representation, allowing for flexible dependence across multipoles while preserving isotropy.

Based on joint works with L. Cristofaro, N. Leonenko, D. Marinucci.

Aiming to construct new stochastic growth models required in a variety of applications, we consider the drift transformations of one-dimensional diffusion processes based on Doob h-transforms. This approach leads to transformed diffusions whose transition density is explicitly expressed through a special time-varying product form. After describing the main features of the method, we focus on the moments of the transformed process, as well as on its local time, the related asymptotic moments and their Laplace transform-based representation. For the transformed diffusion process, we also discuss (i) the first-passage time problem, with attention to the general Laplace transform of the density and their moments, and (ii) the dynamics in the presence of catastrophes. Such theoretical characteristics are then applied to the diffusion obtained from the lognormal process, this being finalized to propose new suitable growth models.

In the past 100 years statisticians made use of different notions of weak dependence for stationary time series.
Beyond ergodicity which essentially ensures the consistency of statistical procedures various notions adapted to prove limit theorems in distributions similar to those known under independence, the most natural is the central limit theorem holding with a square root n normalization. One main name for those developments was Rosenblatt.
Then many authors introduced various notions of mixing, which are related in my 1994 monograph. I realized that they are not always relevant even for simple autoregressive models as reported by Rosenblatt himself and Andrews in 1984. Then with Louhichi we introduced weak dependence notions easy to check and widely adapted to statistical modeling in 1996. After this from 2005 Wu introduced a highly performing notion leading to strong results. Anyway the talks is devoted to develop qualities and limits of all those notions which is an area with many recent developments.

We introduce a wide class of sticky Brownian motions (not necessarily Markovian) and study their boundary occupation time through different notions of holding times. This class consists of all processes obtained from various symbols of subordinators, say Φ. Our discussion focuses on the infinite activity case and, as a reference model, we consider the stable subordinator corresponding to Φ(λ) = λ^α. Sticky diffusion processes on bounded domains can spend a finite time (with a finite mean) on the lower-dimensional space given by the boundary. Once the process hits the boundary, it restarts either instantaneously or after a random period of time. While on the boundary, it can stay or move according to dynamics that differ from those in the interior. Such processes may be characterized by a time derivative appearing in the boundary condition of the governing problem. We restrict our attention to static behaviour without lateral diffusion (i.e., the boundary trace process is a pure jump process). We use suitable time-changes in order to describe fractional sticky conditions and the associated boundary behaviours. We show that fractional boundary value problems (involving fractional dynamic boundary conditions) lead to sticky diffusions, which are strong Markov in the interior and spend a finite time (with an infinite mean) on the boundary. Such behaviour can be interpreted as a trapping effect from a macroscopic point of view.

In this talk, we discuss several forms of scale invariance in continuous and discrete stochastic models. First, we consider stochastic self-similarity (multifractality) of stochastic processes based on scaling of finite-dimensional distributions. We show some general properties of these processes and introduce new examples inspired by the idea of Lamperti transformation.

We then turn to non-negative integer-valued processes, where scaling is defined through thinning operations associated with continuous-time Markov branching processes. This leads to discrete analogues of (stochastic) self-similarity, with time-changed compound Poisson processes providing a natural class of examples. We conclude with ongoing work on related notions for point processes. This research is partly motivated by the so-called Taylor's law, an empirical phenomenon describing a power-law relationship between sample mean and variance.

A time continuous model of the simplest chirp signal observed against the background of strongly or weakly dependent stationary Gaussian noise is considered. For the specified statistical model, strong consistency and asymptotic normality of periodogram-like estimates of unknown amplitude, angular frequency, and chirp rate are obtained.

Recently, Tianshi Lu described second-order mean-square continuous isotropic (resp., reflexive) tangential random fields on a sphere, that are invariant with respect to the special orthogonal (resp., complete orthogonal) group. A careful analysis of his proof shows its dependence on a fundamental fact: both isotropic and reflexive tangential random fields are invariant random cross-sections of the same multiplicity free homogeneous vector bundle over a sphere. The search of similar bundles over other manifolds is a challenge. We demonstrate on a couple of examples, how to find such bundles and their invariant random cross-sections over certain classical and exceptional symmetric spaces and strongly isotropy irreducible Riemannian manifolds.

In this talk we try to show how tools from spherical harmonic analysis, stochastic geometry and random fields can be exploited to investigate the structure and properties of deep neural networks. More precisely, we study the geometric properties of random neural networks by investigating the boundary volumes of their excursion sets for different activation functions, as the depth increases. We prove that, for activation functions which are not very regular (e.g., the Heaviside step function), the boundary volumes exhibit fractal behavior, with their Hausdorff dimension monotonically increasing with the depth. On the other hand, for activations which are more regular (e.g., ReLU, logistic and tanh), as the depth increases, the expected boundary volumes can either converge to zero, remain constant or diverge exponentially, depending on a single spectral parameter which can be easily computed. We also discuss quantitative central limit theorems for the excursion volumes and provide simulations to illustrate the theoretical results.

Based on joint works with Simmaco Di Lillo, Leonardo Maini, Michele Salvi and Stefano Vigogna.

This talk is based on our common results with Luisa Beghin and Alessandro de Gregorio.

The Hadamard fractional Brownian motion is a Gaussian process which shares some properties with standard Brownian motion (such as the one-dimensional distribution). However, it also resembles the fractional Brownian motion in many other features as, for instance, selfsimilarity, long/short memory property, Wiener-integral representation. The logarithmic kernel in the Hadamard fractional Brownian motion represents a very specific and interesting aspect of this process. Our aim here is to analyze some properties of the process’ trajectories (i.e. Hölder continuity, quasi-helix behavior, power variation, local nondeterminism) that are both interesting on their own and serve as a basis for the Wiener integration with respect to it. The respective integration is quite well developed, and the inverse representation is also constructed. We apply the derived “multiplicative Sonine pairs” to the treatment of the Reproducing Kernel Hilbert Space of the Hadamard fractional Brownian motion, and, as a result, we establish a law of iterated logarithm.

This talk discusses the existence and structure of isotropic and anisotropic spherical random fields. We introduce several classes of anisotropic random fields and present a geometric–spectral classification of second-order anisotropy on the sphere. This framework enables a description of feasible isotropic and anisotropic models and their structural properties. We also examine a range of composite transformations of isotropic random fields and investigate how these transformations generate and characterise anisotropy. The results provide new insights into the interplay between geometry, spectral representations, and anisotropic behaviour of random fields on the sphere.

The talk is based on joint results with C.Durastanti (Sapienza Universita di Roma, Italy) and S. Khan (La Trobe University, Australia).

Deterministic models of continuum mechanics tackled by boundary value problems may be inadequate for various reasons, especially in multiscale problems. Viewed from the standpoint of random microstructures, probabilistic models such as stochastic partial differential equations and stochastic finite elements should naturally involve tensor-valued random fields (TRF) of ranks 1…4 with generally anisotropic realizations. Current developments and some open problems are discussed in the setting of wide-sense homogeneous and isotropic, second-order, mean-square continuous TRFs on mesoscales [1,2,3].

When there is interest in TRFs of constitutive properties (e.g., conductivity, stiffness, damage), experiments can be used to determine/calibrate the correlation functions. Classical homogenization theory predicts that Dirichlet and Neumann boundary value problems yield identical effective tensors only in the representative volume element limit of a deterministic continuum, whereas for smaller domains the two responses generally differ. Using a probabilistic description, we define a coordinate-invariant anisotropy measure based on the Hellinger distance between the probability distributions associated with Dirichlet stiffness and the inverse Neumann tensor [4].

Fracture of all materials and structures – both man-made and natural – critically depends on the stress field near a crack tip. Here, we demonstrate an application of TRFs in assessing the mode III stress intensity factor in a random material microstructure [5].

[1] M. Ostoja-Starzewski, S. Kale, P. Karimi, A. Malyarenko, B. Raghavan, S.I. Ranganathan, and J. Zhang, Scaling to RVE in random media, Adv. Appl. Mech. 49, 111-211, 2016.
[2] A. Malyarenko and M. Ostoja-Starzewski, Tensor-Valued Random Fields for Continuum Physics, Cambridge University Press, 2019.
[3] A. Malyarenko, M. Ostoja-Starzewski, and A. Amiri-Hezaveh, Random Fields of Piezoelectricity and Piezomagnetism, Springer, 2020.
[4] M. Ostoja-Starzewski, A probabilistic measure of anisotropy and SVE-to-RVE scaling, Probab. Eng. Mech. 85, 103979, 2026.
[5] Y.S. Jetti and M. Ostoja-Starzewski, Scale-dependent K_III in composites: a tensor random field approach, Probab. Eng. Mech. 82, 103859-1-10, 2025.

We introduce a fractional Bessel process with constant drift, obtained by time-changing the classical Bessel process through the inverse of an independent stable subordinator. This endows the process with subdiffusive dynamics and long-range dependence. Our main result is an explicit spectral representation of the transition density, from which we derive a strong solution of the associated fractional Cauchy problem, show that the stationary distribution remains the gamma law of the non-fractional case, and obtain a closed-form correlation structure, which is new even for the classical Bessel process. We close with an application to queueing theory, where the process arises as the heavy-traffic limit of the unfinished work in a two-queue polling system with random server interruptions.

In this talk, some recent results obtained jointly with Nikolai on fractional risk models will be presented. In particular, we will focus on two generalized fractional risk models that incorporate the random occurrence of stochastic premiums modelled through fractional Poisson processes: the Generalized Fractional Risk (GFR) model and the Generalized Fractional Compound Risk (GFCR) model.

For both models, we derive exact analytical expressions for the mean and covariance functions, highlighting the long-range dependence induced by the underlying inverse stable subordinator. Under a symmetric setting in which claims and premiums have identical intensities and exponentially distributed sizes, we establish a stochastic comparison result showing that the finite-time ruin probability of the GFCR process is strictly bounded above by that of the classical fractional compound risk model.

Finally, we develop a Monte Carlo simulation algorithm based on the path-wise inversion of an α-stable subordinator to generate sample trajectories of the surplus processes and estimate their finite-time ruin probabilities. The numerical approximations provide further support for, and validation of, the theoretical results.

Based on joint work with Dmitri Finkelshtein, Anatoliy Malyarenko and Yuliya Mishura.

We study several entropy measures for the Poisson distribution, including the Shannon, Rényi, Tsallis, Sharma–Mittal, and generalized Rényi entropies. A natural question is how these quantities depend on the Poisson intensity parameter. For the Shannon, Rényi, Tsallis, and Sharma–Mittal entropies, the behavior is “normal”: the entropy increases with the intensity. In contrast, the generalized Rényi entropies may exhibit “anomalous” non-monotone behavior for certain parameter values. We also discuss their asymptotic behavior for large intensity and present related upper and lower bounds. The results illustrate that different notions of entropy, although closely related, may behave quite differently even for such a classical distribution as the Poisson law.

This paper presents asymptotic theory of integral functionals of nonlinear spatiotemporal random fields in the double spectral domain. Specifically, under some invariance properties, we work in the temporal continuous, and spatial pure point spectral domains. Suitable series expansions in terms of independent random variables are obtained for the limit non-gaussian random variable. Its properties are analyzed, including infinitely divisible property. These results are applied to the morphological analysis of random neural networks.

Acknowledgements. This work has been supported in part by projects MCIN/ AEI/PID2022-142900NB-I00, MCIN/ AEI/PID2025-171143NB-I00, and CEX2020-001105-M MCIN/AEI/10.13039/501100011033.

The talk discusses different fractional extensions of the Poisson process and generalized counting processes obtained by introducing time-change represented by the inverse to the sums of stable and tempered stable subordinators. The governing equations for probability distributions and probability generating functions are stated. The equations involve fractional derivatives of different orders in the form of a generalized telegraph-type operator in time. Closed form expressions for probability distributions and probability generating functions are also provided for several considered models. The talk is based on the joint paper with A. Storozhuk.

In recent times, several rigorous results on simple random exchange models were proved. In this talk, after a short review of the literature, I focus on a subset of results in a recent paper in which we discuss various limits of a simple random exchange model that can be used for the distribution of wealth. We start from a discrete state space - discrete time version of this model and, under suitable scaling, we show its functional convergence to a continuous space - discrete time model. Then, we show a thermodynamic limit of the empirical distribution to the solution of a kinetic equation of Boltzmann type.

Lorentz processes are presented in two different settings. Both cases are characterized by infinite expectation of the free-flight times, contrary to what happens in the classical Gallavotti-Spohn models. Under a suitable Boltzmann-Grad type scaling limit, they converge to non-Markovian random-flight processes with superdiffusive behavior. A further scaling limit yields another non Markovian process, i.e., a superdiffusion obtained by a suitable time-change of Brownian motion. Using a technique based on mixtures of Feller semigroups, the governing equations for the random flights and anomalous diffusion are obtained, which represent a non-local counterpart for the linear-Boltzmann and diffusion equations arising in the classical theory.

Papers:
https://www.sciencedirect.com/science/article/pii/S0304414922000783
https://arxiv.org/abs/2507.02796

This is joint work with Bertram Duering (Warwick), Lorenzo Facciaroni (Rome), Nicos Georgiou (Sussex), Sara Merino Aceituno (Vienna), Costantino Ricciuti (Rome), Bruno Toaldo (Turin)

We investigate the monotonicity of the Mittag-Leffler function E_α with respect to the fractional parameter α via the convex ordering properties for related random variables. In particular, we show that the mapping α ↦ E_α(Γ(1+α)x) decreases on (0,1) for all x ∈ ℝ^*, that the mapping α ↦ E_α(x^α) decreases on (0,2) for all x > 0, and that the mapping α ↦ E_α(-x^α) decreases on (0,1) for all x ≥ 1 but not for all x > 0. Analogous results are presented for the two parameter Mittag-Leffler functions E_{α, β} with β ≥ α, with an emphasis on the extremal case β = α. Time permitting, we shall discuss several applications of these results to Abelian integral equations and subdiffusions.

This is a joint work with Rui Ferreira (Porto).

We study a stochastic Susceptible–Infected–Recovered–Vaccinated (SIRV) epidemic model with vaccination and non-permanent immunity, in a population of non-constant size bounded by a carrying capacity. To capture the irregular, environment-driven pace of an epidemic, the transmission rate is perturbed additively by a time-changed Lévy noise: a conditional Brownian motion together with a doubly stochastic Poisson random field, within the martingale-random-field framework of Di Nunno and Sjursen. A random clock Λ modulates both the continuous fluctuations and the intensity of the jumps, the latter accounting for rare, superspreader-type shocks to contact behaviour. For this system we establish existence and uniqueness of a global positive solution, and derive sufficient conditions for extinction and persistence in mean of the disease, expressed through the model parameters and the characteristics of the time-change process. The theoretical results are illustrated by numerical simulations.

In this talk, we consider the Multivariate GARCH (MGARCH) models, which play a central role in modeling time-varying second-order structures in vector financial time series. While stationarity conditions and basic moment properties of BEKK models are well established, substantially less is known about their unconditional higher-order structure. We derive closed-form expressions for the unconditional cumulants of all orders for the MGARCH-BEKK(1,1,1) model, with emphasis on the fourth-order cumulant of the conditional covariance matrix process. The derivation relies on vectorization and Kronecker operator calculus. These results lead to explicit formulae for unconditional skewness and kurtosis of the return vector, in particular for some non-Gaussian family of distributions. A second major contribution is the development of a spectral representation for the BEKK conditional covariance process, obtained via a state space formulation for the conditional covariance matrix and its innovation term. The spectral density of the conditional covariance matrix function in closed form is given. Some particular attention will be paid to the skewness and kurtosis of the model.

In this talk we will present a generalization of the Fox H-function called Fox-Barnes J-function. Like the Fox H-function, it is defined as a contour integral in the complex plane, but instead of an integrand given by a ratio of products of gamma functions involving several parameters, we use a ratio of products of double gamma functions. We study the conditions for its existence and how to choose a contour of integration based on the involved parameters. We discuss how the Fox H-function appears as a particular case and prove some properties of the Fox-Barnes J-function. As an application, we show how the Laplace transform of the Kilbas-Saigo function can be conveniently written in terms of the Fox-Barnes J-function, even in cases where the usual series representation is not convergent. The motivation for this problem comes from the studies about stochastic models governed by stretched fractional dynamics, where in a generalized renewal processes modelled by the Kilbas-Saigo function, the parameter regime that ensures finite expected value interarrival times leads to a formally divergent Laplace series. This makes it necessary to go beyond classical series-based transforms and use of Fox-Barnes J-functions.

Previous Workshops in this series:

• Modern Trends in Probability Theory and Mathematical Statistics III (1 December 2020)
• Modern Trends in Probability Theory and Mathematical Statistics II (27 February 2012)
• Modern Trends in Probability Theory and Mathematical Statistics I (27 September 2011)
• Workshop on Long-Range Dependence: From Fractional Calculus to Financial Applications (7-11 September 2009)