In several applications, it is necessary to provide a reflection of a stochastic process defined by means of a suitable equation, as for instance a stochastic differential equation driven by an additive Gaussian noise. These reflection problems can be solved by rewriting the SDE in integral form and then using in a proper way the Skorokhod reflection map and its properties, in particular the Lipschitz condition with respect to the uniform norm. The latter strategy clearly works also in the deterministic setting. Furthermore, a slight modification of this argument allows us to extend it to the case of Volterra integral equations with a continuous driver. In this way, one can extend such reflection problems also to the setting, for instance, of time-fractional differential equations with a Caputo derivative. This is not the case, however, for time-fractional differential equations with the Riemann-Liouville derivative, since the initial condition, which is dictated on a fractional integral, becomes a power-type singular driver in the integral formulation. With this in mind, we study reflection problems for both deterministic and stochastic (convlution-type) Volterra equations with weakly (power-type) singular drivers. In particular, in this talk we will prove an existence and uniqueness result for both the deterministic and the stochastic setting: to get it, we will first investigate the behaviour of convolution integral with respect to spaces of weakly singular functions and their modulus of continuity, together with the behaviour of the Skorokhod reflection map with respect to weakly singular functions.

One-dimensional reflected Brownian motion and related one-dimensional models, such as skew-Brownian motion are rather well understood. Two-dimensional and higher dimensional versions appear naturally in the limit theorems in applied probability, for example, queueing theory. I will review some recent progress on the question of pathwise (strong) uniqueness for obliquely reflected Brownian motion in a quadrant.

We indicate a point in time at which the notion of "convergence in law of random elements" became a synonym of "weak convergence of their distributions". The competitive approach, consisting of coupling and due to Hammersley (1952) and Skorokhod (1956) ("the a.s. Skorokhod representation") for many years had been considered a rather complementary tool.

We show that Skorokhod was right: the latter approach is the proper one when we leave the safe area of metric spaces and consider random elements in a larger class of so-called submetric spaces. Our claim is supported by more than 300 examples spread out across the SPDE literature.

After a brief overview on copies-based estimation in stochastic differential equations (SDEs), the talk will focus on the following topic: the least squares (LS) estimation of the drift function of a SDE driven by the fractional Brownian motion of Hurst parameter H > 1/2. A parametric estimator, and a projection LS nonparametric estimator, will be presented. However, when H≠1/2, the solution of the SDE is not a semi-martingale, and the natural extension of the Ito integral involved in the definition of the estimators - the Skorokhod integral - is not computable. So, for the parametric estimator, some statistical properties of a computable approximation defined as the fixed point of a map constructed from the well-known relationship between the pathwise integral and the Skorokhod integral will be presented.

We start with additive stochastic sequence that is based on the sequence of iid random variables and has the coefficients that allow for this stochastic sequence to be dependent on the past. For such a sequence, we formulate the conditions of the weak convergence to some limit process in terms of coefficients and of characteristic function of any basic random variable. We adapt the general conditions to the case where the limit process is Gaussian.
Then we go to the multiplicative scheme in order to get the a.s. positive limit process that can model the asset price on the financial market. So, we assume that all multipliers in the prelimit multiplicative scheme are positive, and this imposes additional restrictions on the coefficients, and in addition, we consider only Bernoulli basic random variables. The next goal is to apply these general results to the case, where the limit processes in the additive scheme are fractional Brownian motion (fBm) and Riemann-Liouville fBm.
In the case of the limiting fBm we consider the prelimit processes that are constructed regarding to Cholesky decomposition of the covariance function of fBm, and in both cases (fractional Brownian motion and Riemann-Liouville fBm) we were lucky in the sense that such coefficients are suitable also for the multiplicative scheme. Our proofs require deep study of the properties of the Cholesky decomposition for the covariance matrix of fBm.

For stochastic measures, we assume only sigma-additivity in probability and continuity of the paths. A symmetric integral of random functions with respect to stochastic measures is defined, and an ordinary stochastic differential equation with such an integral is considered. The averaging principle and stability of the solution are studied. Additionally, the transport equation driven by a stochastic measure is considered.

Limit theorems for stochastic processes are often synonymous with Skorokhod space and Skorokhod’s J1 topology. In recent times, this has been complemented by a surge of interest in Skorokhod’s coarser M1 topology and related constructions that aim for higher flexibility while retaining structure. This, of course, comes with new challenges concerning whether or not fundamental operations are preserved in the limit. With this talk, I hope to shed some interesting light on these matters. In particular, I will address some new observations about the tightness of martingales and the continuity properties of Itô integration in the M1 topology. To make matters more concrete, I will also discuss applications to scaling limits for models of anomalous diffusion. The talk is based on joint work with Fabrice Wunderlich.

Langevin dynamics is a well-known model with numerous applications across various fields of science—from physics to machine learning. In recent years, it has become a core component of diffusion models in generative AI, which are used to generate images and videos. However, despite its successful applications, such models suffer from slow convergence, dictated by the ergodic properties of the underlying Markov process. In this work, we study a discretization of the Langevin stochastic differential equation (SDE), which results in a discrete-time Markov chain commonly used in generative algorithms. We demonstrate how discretization impacts convergence, derive convergence rates, and evaluate the effect of high dimensionality on convergence.

We consider subcritical Galton-Watson branching processes with immigration, where the offspring distributions are determined by an iid random environment. Assuming that the branching mechanism is subcritical thebprocess has a unique stationary distribution under weak conditions on the immigration. We are interested in the tail behavior of the stationary distribution. We consider the scenarios when the tail is determined by the offspring distribution, and when it is determined by the immigration distribution. In the first case we use Goldie's implicit renewal theory. In both cases we show that under general assumptions the stationary distribution has a regularly varying tail.
Part of the talk is joint work with Bojan Basrak (Zagreb).

In this talk we explore multinomial and r-variants of several classic objects in combinatorial probability and branching processes, including random recursive trees, Hoppe trees, random set partitions and compositions. Just as classical combinatorial numbers, such as Stirling, Eulerian, and Lah numbers, play a key role in characterizing the distributions of these processes, their r-analogues also emerge naturally in the exact distributional formulas for the multinomial and r-variants.
The talk is based on a recent joint work with Z. Kabluchko (Münster), A. Iksanov (Kyiv) and V. Wachtel (Bielefeld).

In my talk, I will consider the phenomenon of explosion in general (Crump-Mode-Jagers) branching processes, which refers to the event where an infinite number of individuals are born in finite time. In a critical setting where the expected number of immediate offspring per individual is exactly 1, whether or not explosion occurs depends on the fine properties of the reproduction point process. I will review some known results and explain recent results in this are. In particular, I present a necessary and sufficient condition in the case where the reproduction point process is Poissonian.
The talk is based on joint work with Gerold Alsmeyer (Münster), Konrad Kolesko (Wrocław), and Jakob Stonner (Gießen).

In this talk we consider weak convergence of a semi-Markov process in Euclidean space to a diffusion process under superimposed conditions on the local characteristics of the semi-Markov process. As a result, we prove the weak convergence of a semi-Markov process under the conditions of diffusion approximation with a small series parameter ε → 0 (ε > 0) to a diffusion process, determined by its limit operator, at the same time, the diffusion and drift coefficients are given by the asymptotic formulas of the Poisson approximation conditions.
The talk is based on a joint work with I. Malyk (Chernivtsi).

This talk is intended to present some of the latest results on limit and ergodic theorems for perturbed semi-Markov-type processes in connection with the fundamental results in this area of Academician Volodymyr Korolyuk.

In this report, we consider the non-autonomous stochastic models of population dynamics driven by the system of stochastic differential equations with white noise, centered and non-centered Poisson noises. So, we take into account not only ''small'' jumps, corresponding to the centered Poisson measure, but also the ''large'' jumps, corresponding to the non-centered Poisson measure. In the cases of the mutualism model and predator-prey model the coefficients of corresponding systems of stochastic differential equations satisfies neither the local Lipschitz condition nor the linear growth condition. The solution of corresponding stochastic differential equations must be positive because they represent the size of the population. We presented the theorems on the existence and uniqueness of a global, positive solution to the corresponding systems of stochastic differential equations. For considered models we derived sufficient conditions of stochastic permanence, non-persistence in the mean, weak and strong persistence in the mean and extinction of the populations.

The talk introduces a new class of random fields, supCAR fields, which are constructed as superpositions of continuous autoregressive random fields. These supCAR fields possess infinitely divisible marginal distributions. Their second-order properties are characterised by a novel family of covariance functions that allows both short- and long-range spatial dependencies. Functional limit theorems for SupCAR fields are derived under general assumptions. Four limiting scenarios that depend on the marginals of the underlying autoregressive fields and the specifications of the superposition are identified. The obtained limit theorems can be employed for the statistical inference of supCAR fields.
The talk is based on joint results with Prof. N. Leonenko (Cardiff University, UK) and Prof. A. Olenko (La Trobe University, Australia).

My talk is devoted to the relation between Poincaré, (log)Sobolev and concentration inequalities. We discuss "stochastic" versions of Poincaré and (log)Sobolev inequalities, the Herbst argument for deriving the concentration inequality, as well as some examples.

Spherical random fields are very useful for modelling some phenomena in areas such as earth sciences (like, for example, in geophysics and climatology) and cosmology. In fact, the application of statistical methods in cosmology has become increasingly important due to the many experimental data obtained in recent years, and spherical random fields are of particular interest regarding the analysis of Cosmic Microwave Background (CMB).
As well-known, the CMB is a spatially isotropic radiation field spread throughout the visible universe, originated around 14 billion years ago, and it is the main source of information we have about the evolution of the universe. The CMB radiation can be mathematically modelled as an isotropic spherical random field for which there is a spectral representation by means of spherical harmonics.
Our objective is to study the fundamental solutions to fractional hyperbolic diffusion equation in the time variable using the Caputo derivative, and its properties. The exact solutions of the fractional hyperbolic diffusion equation with random data in terms of series expansions of isotropic in space spherical random fields on the unit sphere are derived, and numerical illustration are presented to illustrate the results. Some limit theorems for spatio-temporal random fields have been obtained in [1,2].
This is joint results with J.Vaz (UNICAMP, Brazil) and A. Olenko (La Trope University, Melbourne, Australia) [3,4].

References:
[1] Leonenko, N., Maini, L., Nourdin, I. and Pistolato, F. (2024) Limit theorems for p- domain functionals of stationary Gaussian fields, Electronic Journal of Probability, 29,1-33.
[2] Leonenko, N. and Ruiz-Medina, M.D. (2025) High-Level Moving Excursions for Spatiotemporal Gaussian Random Fields with Long Range Dependence, Journal of Statistical Physics, 192, N2, Paper No. 19, 29pp.
[3] Leonenko, N. and Vaz, J. (2020) Spectral analysis of fractional hyperbolic diffusion equations with random data, Journal of Statistical Physics, 179,155-175.
[4] Leonenko, N., Olenko, A. and Vaz, J. (2024) On fractional spherically restricted hyperbolic diffusion random field, Communications in Nonlinear Science and Numerical Simulation, 131,107866.

Finite mixture models (FMM) naturally arise in statistical analysis of medical and biological data, when observed subjects belong to many sub-populations with different distributions. In the lecture we will discuss parametric and nonparametric techniques of estimation under FMM. Special attention will be paid to the case when the mixing probabilities can vary at different observations.

We study two related stochastic processes driven by Brownian motion: the Cox-Ingersoll-Ross (CIR) process and the Bessel process. We investigate their shared and distinct properties, focusing on time-asymptotic growth rates, distance between the processes in integral norms, and parameter estimation. The squared Bessel process is shown to be a phase transition of the CIR process and can be approximated by a sequence of CIR processes. Differences in stochastic stability are also highlighted, with the Bessel process displaying instability while the CIR process remains ergodic and stable. The talk is based on the common results with K.Ralchenko and S. Kushnirenko.

Multifractional processes extend the concept of fractional Brownian motion by replacing the constant Hurst parameter with a time-varying Hurst function. This allows to model systems with changing dynamic and to modulate the roughness of sample paths over time. The talk introduces a new class of multifractional processes, the Gaussian Haar-based multifractional processes (GHBMP), which is based on the Haar wavelet series representations. The resulting processes cover a significantly broader set of Hurst functions compared to the existing literature, enhancing their suitability for both practical applications and theoretical studies. The theoretical properties of these processes will be discussed. Then, it is demonstrated how the suggested representation of GHBMP can be easily implemented for simulations with various Hurst functions. The proposed model is validated and its applicability is demonstrated, even for Hurst functions exhibiting discontinuous behaviour.
The talk is based on joint results with Prof. A.Ayache (University of Lille, France) and N.Samarakoon (La Trobe University, Australia).

We study the tempered fractional Brownian motion (TFBM) and the tempered fractional Brownian motion of the second kind (TFBMII) - two stochastic processes that modify the power-law kernel in the moving average representation of fractional Brownian motion by introducing exponential tempering. We establish almost sure asymptotic bounds for the growth rates of the trajectories and increments of both processes. These results are then applied to prove the strong consistency of drift parameter estimators in Ornstein-Uhlenbeck processes driven by TFBM and TFBMII. We consider both the least squares estimator based on continuous-time observations and its discretized counterpart.

We continue to explore the scientific contributions of Yuriy Kozachenko. In this talk, we invite you to examine random variables and stochastic processes within the space Fψ(Ω) and to investigate several of their key properties. It has been established that Fψ(Ω) forms a Banach space under an appropriate norm. Moreover, estimations of the tail distributions for random variables in Fψ(Ω) have been derived. A method based on series decomposition is employed to construct an approximating process, referred to as a model, for stochastic processes in this space. We also discuss the rate of convergence of this model to the original process in the uniform norm.

Global Fréchet functional regression has been recently addressed from time correlated bivariate curve data evaluated in a manifold. This paper derives local linear Fréchet functional regression predictors for the same type of correlated curve data in an extrinsic and intrinsic way. The optimality in the approximation of the Fréchet conditional mean by the proposed extrinsic and intrinsic Fréchet functional predictors is then proved, adopting a weighted Fréchet mean approach. The performance of both Fréchet curve predictors is illustrated in the simulation study undertaken, where a comparative study is achieved. A real-data problem is addressed to illustrate the finite-sample properties of these predictors via cross-validation. In this real-data problem, functional predictions of the magnetic field vector are obtained from the time-varying geocentric latitude and longitude of the satellite NASA's MAGSAT spacecraft.

The talk presents bounds for the distributions of suprema for particular classes of sub-Gaussian type random fields. Results stated depend on representations of bounds for increments of the fields in different metrics. Several examples of applications are provided to illustrate the results, in particular, applications to random fields related to stochastic partial differential equations and partial differential equations with random initial conditions.

The talk explores the properties of extremal functionals, particularly suprema and cumulative functionals, of sub-Gaussian stochastic processes. Examples of sub-Gaussian processes are presented, with a focus on fractional Brownian motion. Applications of the obtained results are discussed in the contexts of queueing theory and financial mathematics.

Professor Yuriy Kozachenko was one of the founders of the theory of φ-sub-Gaussian stochastic processes and processes from Orlich spaces. He inspired his students to study such processes and to apply the obtained results to various stochastic processes appearing in physics, finance and insurance, social sciences etc. Among such processes there are φ-sub-Gaussian quasi shot noise processes. Shot noise processes serve as mathematical models of various physical phenomena. They are used in electronics, telecommunications, mesoscopic physics. In the talk we present estimates for distribution of supremum and conditions for sample paths continuity with probability one for φ-sub-Gaussian quasi shot noise processes. We also consider the problem of estimating the probability of exceeding some level by such a process on the finite interval. We are extremely grateful to our teacher Yuriy Kozachenko for inspiration, encouragement and invaluable advice.

A class of multiparameter distributions generated by the distribution of a random variable with independent identically distributed digits of the Q_s-representation is considered. For this class of distributions, a modification of the maximum likelihood method for the estimating of unknown parameters is proposed. We prove that obtained estimates are unbiased, consistent, and efficient.

On the real line, we study self-similar sets, which are arised as sets of subsums of convergent positive series. In certain simple cases, these sets are either nowhere dense fractals or finite unions of closed intervals. However, in more complicated cases, these sets may take the form of Cantorvals—perfect sets with non-empty interior and fractal boundary, which coincides with the closure of the interior. In addition, we compute the Lebesgue measure of the interior and the Hausdorff dimension of the boundary for Cantorvals belonging to a specific family.

The talk is devoted to the study of the asymptotic properties of the Fourier–Stieltjes transforms of a class of generalized infinite Bernoulli convolutions.

Let us consider a set of numbers in the b-ary expansion for which the frequency does not exist for any individual digit. Such numbers is said to be essentially non-normal numbers. In 2005, Albeverio, Pratsiovytyi and Torbin proved that this set has full Hausdorff dimension and it is of second Baire category. This result was extended for various numeral systems with finite alphabet and infinite alphabet. We extend and generalize this result for a large class of Cantor series expansions considering numbers for which frequency does not exist for any block of digits for any length. Furthermore the result still holds for a set of essentially non-normal numbers whose Cantor series digits are sampled along all arithmetic progressions.

TBA

The talk is devoted to the structural, fractal, and differential properties of continuous functions defined in terms of the As -continued fraction representation of real numbers as well as to probability distributions related to functions of this class.

During the talk we shall discuss open problems related to fractal properties of distributions of random variables with independent symbols with different alphabets.

The talk is devoted to the study of sufficient conditions for the faithfulness for the Hausdorff dimension calculation of systems of cylinders generated by Q̃-expansions. Based on the obtained results, fine fractal properties of distributions of random variables with independent Q̃-symbols are studied.

The talk is devoted to the study of conditions for the Hausdorff–Besicovitch dimension preservation on [0, 1] by probability distribution functions of random variables with independent symbols of arithmetic Cantor series expansion.