## Poster session

### Long time fluctuations at critical parameter of Hopf bifurcation

**Michele Aleandri** *LUISS University*

We consider an n-dimensional family of deterministic processes depending on a parameter µ and we suppose that an Hopf bifurcation occurs at a nonhyperbolic equilibrium point at critical value µc. A multiplicative noise depending on a small parameter is added to the deterministic process. We analyse the limit, as the parameter of the noise goes to zero and at the critical value µc, of the stochastic process when we rescale time and space. We prove that the dynamics of the fluctuations can be described by a stochastic differential equation whose coefficients depend on the central manifold at the nonhyperbolic equilibrium point and they are obtained through an averaging principle. Join work with Paolo Dai Pra (Università di Verona).

### A numerical study of a PDE-ODE system with a stochastic dynamical boundary condition: a nonlinear model for sulphation phenomena

**Francesca Arceci** *Università degli studi di Milano*

The problem of degradation of marble stones has been a very important issue that has been observed in the last century. The first cause is due to atmospheric pollutants, in particular the reaction of sulphur dioxide with calcareous surfaces which forms gypsum and black crust. Recently, some deterministic mathematical models have been used to study the evolution of the degradation phenomena (Natalini at al, 2004,2023). In this talk we present a numerical study a PDE-ODE model coupled with a stochastic dynamical boundary condition. This boundary condition is given by a Pearson diffusion process, solution to a Brownian motion-driven stochastic differential equation with a mean reverting drift and a bounded diffusion coefficient. The coefficient of the Pearson process are chosen in order to satisfy the well-posedness and the boundedness of the solution. Since classical numerical approximation schemes for SDEs could not be applied directly due to the non-Lipschitz property on the diffusion coefficient, we have used the Lamperti transform to obtain a new process where the diffusion coefficient is constant. We numerically discretize the latter by adopting a Smooth Slope Truncation Scheme to ensure the monotonicity of the drift coefficient and we finally go back to the initial Pearson process through the inverse function. We show the L2 error analysis obtained via Monte Carlo simulation at the final time between a reference solution and the numerical approximated one. Existence and uniqueness of the mild solution of the random PDE-ODE system are stated by (Maurelli, Morale, Ugolini, 2023) adopting a splitting strategy which allows to deal with the low regularity of the dynamical boundary condition. The solution is given as the sum of two contributions: the first one solves the classic heat equation with the stochastic dynamical boundary condition, while the second one solves a non-linear coupled system with constant boundary conditions. A complete qualitative and numerical analysis of the solution’s behaviour of the non-linear PDE-ODE system with the stochastic process at the boundary is illustrated. In analogy to the theoretical estimations, we also build a numerical scheme based on the above splitting of the solution and we compare the results. Numerical convergence and stability are discussed. This is a joint work with Daniela Morale, Stefania Ugolini (University of Milano), Mario Maurelli (University of Pisa), (Francesco de Vecchi, University of Pavia).

### Infinite-dimensional Gaussian change of variables' formula

**Claudio Asci** *Università degli Studi di Trieste*

In this paper, we study the Banach space $\ell _{\infty }$ of the bounded real sequences, and a measure $N(a,\Gamma )$ over $\left( \QTR{bf}{R}^{\infty },\QTR{cal}{B}^{\infty }\right) $ analogous to the finite-dimensional Gaussian law. The main result of our paper is a change of variables' formula for the integration, with respect to $N(a,\Gamma )$, of the measurable real functions on $\left( E_{\infty },\QTR{cal}{B}^{\infty }\left( E_{\infty }\right) \right) $, where $E_{\infty }$ is the separable Banach space of the convergent real sequences. This change of variables is given by some $\left( m,\sigma \right) $ functions, defined over a subset of $E_{\infty }$, with values on $E_{\infty }$, with properties that generalize the analogous ones of the finite-dimensional diffeomorphisms.

### On continuity of state-dependent utility

**Edoardo Berton** *Università Cattolica del Sacro Cuore*

State dependent expected utilities for a general Savage's state space were shown in Wakker and Zank (1999) to admit a numerical representation in the form of the integral of a state-dependent utility, as soon as pointwise continuity of the preference ordering is assumed. In this paper we prove that such a state-dependent function inherits pointwise continuity from the preference ordering, providing in this way a positive answer to a conjecture posed in the aforementioned seminal work. We further apply this result to obtain an explicit representation of conditional Chisini means, studied in Doldi and Maggis (2023), in the form of conditional certainty equivalent. Join work with Maggis Marco (Università degli Studi di Milano) and Doldi Alessandro (University of Florence).

### Interval scale classification: A method for adapting ordinal performance metrics

**Giulia Binotto** *Universitat Autònoma de Barcelona*

In the field of supervised machine learning, the accurate assessment of classification models holds great importance in both performance evaluation and strategic selection of models. We focus on ordinal classification, with a special interest in interval scale measurement. Traditional metrics fail to capture the inherent ordinal structure of data, and a necessity arises for new performance metrics to address this limitation. After exploring the established metrics designed for ordinal classification, we extend the results to interval scale classification, where the length of the intervals, not only their order, assumes significance. Our idea is to offer a comprehensive evaluation framework for this scenario. In this direction, we introduce a comprehensive methodology for adapting any ordinal metric defined on the confusion matrix to the interval scale, and we present the properties of this new performance evaluation. Finally, we deal with the challenging classification with unbounded rightmost intervals, which further enhances the applicability of the proposed metrics.

### First passage times estimates for time-changed Brownian motion: paths simulation and hazard-rate methods

** Luigia Caputo ** *Università degli Studi di Napoli Federico II*

Anomalous diffusions are often observed in real data, that can manifest in asymmetric densities, heavy tails, sharp peaks, different spreading rate.Among such diffusions, we can find some time-changed processes, obtained by composing two independent processes: the outer process and the inverse of an α-stable subordinator. The central role is played by the time-changed Brownian motion (TCBM), that is characterized by continuous sample paths having time periods (of a random duration) in which the process remains trapped. Sometimes this process is also called delayed Brownian motion (BM). For application purposes the first passage time (FPT) of this process is the core of many studies but no closed form results are known. We refer to a subordinated density function of the FPT of the time-changed Brownian motion as the integral of the FPT density of the BM with respect to the density of the inverse stable subordinator. Taking this function into account, our idea is to generalize the hazard rate method (HRM) that we implemented in an R package for FPT simulation of Gaussian diffusions. We propose two different generalizations of the HRM by considering subordinated hazard rates. Results are provided in graphical form for different values of the stability order of the subordinator in case of a constant threshold. Comparisons are also made with FPT obtained by simulation of the trajectories of the TCBM, in order to verify the agreement of the two approaches. Join work with Maria Francesca Canfora (Consiglio Nazionale delle Ricerche) and Enrica Pirozzi (Università della Campania Luigi Vanvitelli).

### Separable priors for Gaussian graphical models

**Jack Carter** *Università di Genova *

A detailed examination of the use of separable priors, in particular spike and slab priors, for a Gaussian precision matrix. Specific considerations needed for such priors will be highlighted as well as some best practices for the choice of prior. Particular attention will be made to the issues of scale invariance and positive definiteness of the prior, along with the importance of the diagonal entries of the precision matrix and how to best choose the prior for these diagonals.

### Monte Carlo estimation for Fractional Partial Differential equations

**Daniel Eduardo Cedeño Giron** *Università di Torino*

We consider a class of non-local partial differential equations associated with semi-Markov processes. Based on this probabilistic interpretation, we study numerical approximations, the convergence and its rate via the Central Limit and the Berry-Esseen theorems, and the upper bound of errors between the exact solution and the Monte Carlo approximation. We study, in particular, the semi-Markov processes constructed as a time-change, i.e., a Feller process evaluated at some suitable random time change. We focus on the specific case when the time change is the undershoot of a continuous stochastic process with non-negative, stationary, and independent increments, which is also independent of the Feller process. To improve the accuracy of our simulations, the undershoot is sampled from its density using an effective rejection sampling method rather than random walk approximations. Meanwhile, the Berry-Esseen type bound allows us to choose a desired number of Monte Carlo iterations to control the error.

### Sturm-Liouville systems for the survival probability in first-passage time problems

**Marcus Dahlenburg** *Sapienza Università di Roma*

We derive a Sturm-Liouville system [1] of equations for the exact calculation of the survival probability of symmetric random walkers in the semi-infinite interval [2]. Within the discrete time setting, the Sparre Andersen theorem [3] applies as a boundary condition for the system that is associated with the Wiener-Hopf integral equation obtained from the theory of random walks [4]. We provide the extension to the continuous-time setting [5], so that our derived approach is feasible for both, discrete-time and continuous-time random walks. Our results will be tested against a manageable, but meaningful, example that allows for the application in the non-Markovian framework [6], too. Join work with G. Pagnini.

References

[1] Tricomi, F. G. 1985 Integral Equations. Dover.

[2] Dahlenburg, M. and Pagnini, G. 2023, Sturm-Liouville systems for the survival probability in first-passage time problems. Proc. R. Soc. A 479, 20230485.

[3] Sparre Andersen, E. 1954 On the fluctuations af sums of random variables II. Math. Scand. 2, 195–223.

[4] Bray, A. J., Majumdar S. N., and Schehr G. 2013 Persistence and first-passage properties in nonequilibrium systems. Adv. Phys. 62, 225–361.

[5] Dahlenburg, M. and Pagnini, G. 2022 Exact calculation of the mean first-passage time of continuous time random walks by nonhomogeneous Wiener-Hopf integral equations. J. Phys. A: Math. Theor. 55, 505003.

[6] Hilfer, R. and Anton, L. 1995 Fractional master equations and fractal time random walks. Phys. Rev. E 51, R848.

### Stability and geometric approximation of nonlinear filters

**Eliana Fausti ** *Imperial College London *

Nonlinear filtering consists in finding the best estimate for the true value of a system when only incomplete, noisy observations are available. Since the filtering equations are often very high dimensional, and not tractable from a practical perspective, many applications must rely on approximation methods. Understanding the error of several such approximations remains an open problem. In this talk we will focus on the case where the underlying process is a continuous-time finite state-space Markov chain. Levaraging tools from information geometry, we will show how to obtain a contraction result for the filtering equations, strengthening now-classical results on the stability of filters. This allows us to characterize the error of approximate filters, and to constuct a low-dimensional geometric approximation with a well-behaved, computable error. Based on joint work with S. N. Cohen (University of Oxford).

### New a Priori Estimate for Stochastic 2D Navier-Stokes Equation with Applications to Invariant Measure

**Matteo Ferrari** *Università di Pavia*

The paper deals with the two-dimensional stochastic incompressible Navier-Stokes equation set in a bounded domain with Dirichlet boundary conditions. We consider an additive noise in the form of a cylindrical Wiener process regularized by a term $A^{-\gamma}$, where $A$ is the Stokes operator, and $\gamma\in(1/4,1/2)$. We prove uniqueness, ergodicity, and a strong mixing property for the invariant measure of the Markov semigroup. While previous results require $\gamma > 3/8$, we uncover the range $\gamma \in (1/4, 3/8]$ by adapting the so called Sobolevskii-Kato-Fujita approach to stochastic Navier-Stokes equations. By means of the mild formulation, this method gives a new \textit{a priori} estimate for the trajectories of the solution, which entails H\"older continuity in time and regularity $D\big(A^{\gamma'}\big)$ in space, where $\gamma'<\gamma$.

### Finite State Graphon Mean Field Games: a Label-State Formulation

**Nicola Fraccarolo ***Università di Trento*

In this work we will talk about stochastic games on large graphs, where the players no longer interact with each other symmetrically. To encode this information, the concept of graphon is employed. Graphons are the natural continuum limits for dense interaction matrices. Matematically, a graphon is a symmetric measurable function W : [0,1] → [0,1], with W(u,v) representing the interaction between vertices players u and v. We consider a continuous-time controlled dynamics on finite states: we write the dynamics of the N-player game as a system of stochastic differential equations driven by independent stationary Poisson random measures. Under a set of fairly general assumptions, we derive the existence of both relaxed and non-relaxed controls, while uniqueness is proven under the Lasry-Lions monotonicity conditions. This talk is based on joint research with Luca Di Persio (University of Verona, Italy) and Francesco Giuseppe Cordoni (University of Trento, Italy).

### Adaptive Elastic-net estimation for ergodic diffusion processes

**Dario Frisardi ** *Sapienza Università di Roma*

Elastic-net regularization has emerged as a powerful tool in parameter estimation addressing challenges posed by multicollinearity and high-dimensional data. It combines both the strengths of Lasso, allowing for variable selection, and Ridge, penalizing the magnitude of coefficients to control multicollinearity. Adaptive weights are considered in the L1-penalty to ensure that the variable selection procedure exhibits the so-called oracle properties. Recently, regularization method have been applied to multidimensional diffusion processes, allowing for an expanded understanding of stochastic processes in the field of statistics. In this study, we investigate the Adaptive Elastic-net regularization for the estimation of parameters governing drift and diffusion coefficients for ergodic diffusion processes. We examined the desirable properties of this estimator such as consistency, selection consistency and asymptotic normality. Moreover, we conduct simulations and apply this method to real data to provide empirical evidence of its applicability. This research represents joint work with A. De Gregorio, F. Iafrate and S. M. Iacus.

### Well-Posedness and stationary solutions of McKean-Vlasov (S)PDEs

**Martin Kolodziejczyk** *Heriot-Watt University*

Many systems in the applied sciences are made of a large number of particles. One is often not interested in the detailed behaviour of each particle but rather in the collective behaviour of the group. An established methodology in statistical mechanics and kinetic theory allows one to study the limit as the number of particles N tends to infinity and to obtain a PDE for the evolution of the density of particles. The limiting PDE is a non-linear equation, where, in the instance of interest to us, the structure of the non-linearity is commonly referred to as the McKean-Vlasov non-linearity. It is well-known that, even when the particle system has a unique invariant measure (stationary solution), the limiting PDE very often displays a phase transition: for certain choices of (coefficients and) parameter values, the PDE has a unique stationary solution, but as the value of the parameter varies multiple stationary states appear. We join this stream of literature and consider a specific instance of a McKean-Vlasov type equation, namely the Kuramoto model on the torus perturbed by a symmetric double-well potential, and show that this PDE undergoes the type of phase transition just described, as the diffusion coefficient is varied. After that, we consider a rather general class of McKean-Vlasov PDEs on the torus (which includes both the original Kuramoto model and the Kuramoto model in double well potential of part one) perturbed by an (strong enough) infinite-dimensional additive noise and focus on their well-posedness and long-time behaviour. Lastly, we address the question about whether it is possible to obtain McKean-Vlasov SPDEs with additive noise from particle systems. We explain how to tackle this problem by studying limits of weighted particle systems, in a framework introduced by Kurtz and collaborators. This is a joint work with L. Angeli, J. Barrè, D. Crisan and M. Ottobre

### On the Construction of Stationary Processes with Generalized Bernoulli Process

**Jeonghwa Lee ** *University of North Carolina Wilmington*

Long-range dependence refers to a phenomenon where there is a strong correlation among a sequence of random variables such that its covariance function is not summable, or decays with a power-law, and it is measured by the Hurst index. In Lee (2021), a generalized Bernoulli process (GBP) was developed which is a stationary binary sequence that can have long-range dependence. In the first part of the talk, GBP will be introduced with its asymptotic properties and its connection to the fractional Poisson process. In the second part of the talk, we will discuss the construction methods of stationary processes with GBP. GBP is extended to a finite-state stationary process in which different states can have different Hurst parameters. More generally, a construction method of a stationary process with any one-dimensional marginal distribution and a given convex, decreasing covariance function is developed. The latter method is easily extended to construct a multivariate stationary process and random field with any one-dimensional marginal distribution and a given covariance function, and simulations of such models will be examined.

### Non-negativity and zero isolation for generalized mixtures of densities

**Giulia Lombardi ** *Università degli Studi di Trento*

In the literature, finite mixture models are described as linear combinations of probability distribution functions having the form $\displaystyle f(x) = \Lambda \sum_{i=1}^n w_i f_i(x)$, $x \in \mathbb{R}$, where $w_i$ are positive weights, $\Lambda$ is a suitable normalizing constant and $f_i(x)$ are given probability density functions. The fact that $f(x)$ is a probability density function follows naturally in this setting. However, what happens if we remove the condition that the weights $w_i$ must be non-negative? In other words, what if some of the weights can be negative? \\ This question is addressed by proposing a novel approach that utilizes a finite sequence of function known as generalized Budan-Fourier (GBF) sequence to accurately identify the zero-crossings of the function $f(x)$. By analyzing these zero-crossings, we can determine whether the resulting function is indeed a probability density function for any given choice of the weights. Theoretical justification for the functioning of the algorithm and demonstration of its effectiveness through various examples will be provided. Furthermore, the potential applications of the GBF approach will be explored by providing a particular focus on generalized Gaussian mixture densities. In conclusion, this work contributes to the development of robust and versatile techniques for constructing probability density functions from finite mixtures, even in scenarios where some of the weights may be negative. This extension broadens the applicability of finite mixture models and paves the way for more flexible and expressive modeling of complex statistical distributions. Joint work with Hanzon Bernard (University College Cork) and Bonaccorsi Stefano (Università degli Studi di Trento).

### Strong uniqueness for degenerate SDEs with rough coefficients: recent developments and open problems

**Giacomo Lucertini** *Università di Bologna *

We consider a class of kinetic stochastic differential equations with low regularity coefficients: these are assumed to be α-Hölder continuous in space (and velocity) and only measurable in time. The well-posedness of this kind of equations have been widely studied: recently, have been established some thresholds for α for the weak and strong uniqueness of kinetic SDEs, together with counterexamples, leaving as an open problem the pathwise uniqueness when α ∈ [1/3, 2/3]. We investigate this case using a PDE technique: the approach is based on the study of the regularity of solutions for the backward Kolmogorov equation associated to the SDE. The results are based on a joint project with Stephane Menozzi and Stefano Pagliarani.

### Dynamic local convergence and dynamic giant process

**Marta Milewska** *Centrum Wiskunde & Informatica*

Local convergence has become a very important tool in sparse random graph theory, as many random graph properties turn out to be determined by the local limit. For instance, the asymptotic number of spanning trees and the partition function of the Ising model are computable in terms of the local limit. So far, local convergence has been mostly investigated in the context of static graphs. However, next to some other recent work on local convergence of dynamic graphs in, in our recent paper we show how we can naturally extend this notion to the dynamic setting by applying classic theory on the convergence of stochastic processes. Thanks to this, the power of local convergence technique can also be applied to tackle the properties of dynamic graphs. As one possible example, we describe how thanks to the dynamic local convergence we can describe membership in the giant component in a dynamic graph.

### Optimal Stopping of Branching Diffusion Processes

**Antonio Ocello** *Ecole Polytechnique*

In this talk, we will explore an optimal stopping problem for branching diffusion processes. It consists in looking for optimal stopping lines, a type of stopping time that maintains the branching structure of the processes under analysis. By using a dynamic programming approach, we characterize the value function for a multiplicative cost that depends on the particle’s label. We reduce the problem’s dimensionality by setting a branching property and defining the problem in a finite-dimensional context. Within this framework, we focus on the value function, establishing polynomial growth and local Lipschitz properties, together with an innovative dynamic programming principle. This outcome leads to an analytical characterization with the help of a nonlinear elliptic PDE. We conclude by showing that the value function serves as the unique viscosity solution for this PDE, generalizing the comparison principle to this setting. Joint work with Kharroubi Idris (LPSM, Sorbonne Université).

### Fastest-Mixing Markov Chain

**Sam Olesker-Taylor** *University of Warwick*

A standard objective is to sample uniformly from a set V via a Markov chain which is only allowed to make transitions between a given collection E of pairs of V. Eg, perhaps only ‘local’ updates are permitted, in some sense. Our goal is to design a Markov chain on the graph G = (V, E) with mixing time as fast as possible, in order to make the sampling algorithm computationally efficient. This is termed the “fastest-mixing Markov chain” question, and was introduced by Boyd, Diaconis and co-authors in the mid ’00s. A lower bound on the fastest mixing time had been determined by Roch (’05): “no mixing time can be faster than…”. We prove an upper bound, establishing a Cheeger-type inequality bounding the fastest mixing time with the vertex conductance. We also consider chains with “almost-uniform” invariant distribution π: given ε > 0, we require π(x) ≥ (1−ε)/|V| for all x ∈ V. We construct a Markov chain on G with mixing time at most (diam G)² (log |V|)/ε; this bound, somewhat remarkably, does not depend on the local structure of the graph. The construction also gives a continuous-time Markov chain on G with exactly-uniform invariant distribution and average jump-rate 1 which has mixing time at most (diam G)² log |V|. This is joint work with Luca Zanetti (Bath).

### Fractional integrals of stochastic processes for neuronal models and simulations

**Enrica Pirozzi ***Università della Campania Luigi Vanvitelli*

Insiperd by the work of Teka et al. ([1]) we focus on fractional neuronal models, based on frac- tional stochastic dierential equations with the fractional Caputo-derivative in place of the classical derivative. In such a way a fractional version of the stochastic Leake Integrate-and-Fire (sLIF) model can be considered. The adoption of this kind of models shows some advantages among them the possibility to describe dynamics on dierent time-scales and also to convoy some memory eects. Here, we consider more sophisticated models based on a fractional stochastic dierential equations with a stochastic correlated process in place of the classical white noise. The correlated process is considered to model a correlated input to the neuronal dynamics. We follow and generalize previous results as those in [2], [3] and [4]. The solution processes of such fractional dierential equations are fractional integrals of correlated processes, in some cases of fractional correlate processes. On the study of such processes is focused the present contribution. In particular, we provide the expected value and the covariance of the fractional integrals of these processes, specied for dierent choices of the coecient functions of the considered equations. Consequentially, we are able to propose three dierent neuronal models. After a mathematical analysis of the considered fractional equations ([5]), we also investigate possible simulation techniques ([6], [7],[8]) in order to obtain estimates of rst passage times of such processes through a constant level (the neuronal ring threshold) especially useful for modeling the neuronal dynamics.

References

[1] Teka W., Marinov T.M. and Santamaria F., Neuronal Spike Timing Adaptation Described with a Fractional Leaky Integrate-and-Fire Model, PLoS Comput Biol., 10 (2014).

[2] Bazzani A., Bassi G. and Turchetti G., Diusion and memory eects for stochastic processes and fractional Langevin equations, Phys. A Stat. Mech. Appl., 324, 530-550, (2003)

[3] Pirozzi E., Colored noise and a stochastic fractional model for correlated inputs and adaptation in neuronal ring. Biological Cybernetics, 112 (1-2), 2539, (2018)

[4] Ascione G., Pirozzi E., On a stochastic neuronal model integrating correlated inputs, Mathematical Biosciences and Engineering, Volume 16, Issue 5: 5206-5225. doi: 10.3934/mbe.2019260, (2019)

[5] Anh, P.T., Doan, T.S., Huong, P.T., A variation of constant formula for Caputo fractional stochastic dierential equations, Statistics and Probability Letters, Volume 145, Pages 351-358, (2019)

[6] DoanT.S., Huong P.T., Kloeden P.E., Vu A.M., Euler Maruyama scheme for Caputo stochastic fractional dierential equations, Journal of Computational and Applied Mathematics, Volume 380, 112989, (2020)

[7] Abundo M., Pirozzi E., Abundo, M., Pirozzi, E., Fractionally integrated Gauss-Markov pro- cesses and applications, Communications in Nonlinear Science and Numerical Simulation, Volume 101, 2021, 105862, ISSN 1007-5704, (2021)

[8] Pirozzi, E., On the Integration of Fractional Neuronal Dynamics Driven by Correlated Processes, LNCS volume 12013, Computer Aided Systems Theory EUROCAST 2019, pp- 211219, (2019)

### Mean field approximation of a stochastic particle system to reaction-diffusion model for sulphation phenomenon

**Giulia Rui** *Università degli Studi di Milano*

We propose a first stochastic model for the sulphation phenomenon in Cultural Heritage, at the molecular level. We describe the Langevin dynamics via stochastic differential equations of It\^o type. More precisely, we consider a system of stochastic differential equations driven by a family of independent Wiener processes, of first order for the solid molecules and of second order for the fluid ones, coupled with a marked Poisson compound point measure for the chemical reactions. Equations model the individual interactions among molecules, by means of a Lennard Jones type potential and a suitable model for the porosity of the material. The Eulerian description is given via the empirical measures, solution of measure valued stochastic equation. The choice of suitable rescalings for the system is discussed. A specific time-space rescaling for the first order Eulerian system as the number of particles increases is proposed, in order to derive a deterministic continuous porous media dynamics at the macroscale. In the limit we obtain a known deterministic model (Natalini at al., 2004,2023), consisting of a parabolic PDE coupled with an ODE for an external field. This is a joint work with Daniela Morale, Stefania Ugolini (University of Milano).

### Exchangeable measure-valued Pólya urn sequences

**Hristo Sariev** *Sofia University St. Kliment Ohridski*

Measure-valued Pólya urn sequences (MVPS) are a generalization of the observation processes generated by $k$-color Pólya urn models, where the space of colors $\mathbb{X}$ is a complete separable metric space and the urn composition is a finite measure on $\mathbb{X}$. In this talk, we give a complete picture of the class of exchangeable MVPSs. First, we state a representation theorem for the reinforcement measures $R$ of all exchangeable MVPSs, which leads to a characterization result for their directing random measures. In particular, when $\mathbb{X}$ is countable or $R$ is dominated by the initial distribution, then every exchangeable MVPS is a Dirichlet process mixture model over a family of probability distributions with disjoint supports. We further characterize the predictive distributions of exchangeable MVPSs in terms of a ``sufficientness'' postulate and discuss the relation of MVPSs to the class of exchangeable Hoeffding decomposable processes. Importantly, we do not restrict our analysis to balanced MVPSs, in the terminology of k-color urns, but instead show that the only non-balanced exchangeable MVPSs are sequences of i.i.d. random variables. This is joint work with Mladen Savov.

### Exploring and Applying Divergence Measures and Distances in the Analysis of Compositional Data and Their Real-World Applications

**Muhammad Shoaib ***Università di Genova*

Compositional data require particular metrics to define the distance and dissimilarity across probability distributions. This research investigates various divergence measures (especially Kaniadakis’ divergence) and lengths employed in compositional data analysis, with a particular focus on their real-world applications. We explore these measures’ theoretical foundations and practical implications, aiming to discern their effectiveness in capturing dissimilarities within compositional datasets. Through applications to real-world problems, this study seeks to enhance our understanding of the practical utility and performance of these analytical tools in diverse contexts. The findings contribute valuable insights to researchers and practitioners engaged in analyzing and interpreting compositional data across various domains.

### Distribution related to all the samples and extreme events in the Epidemic Type Aftershock Sequence (ETAS) clusters

**Ilaria Spassiani** *Istituto Nazionale di Geofisica e Vulcanologia*

Forecasting seismic events is a challenging goal for seismologists and geophysicists, due to the complex nature of the earthquake phenomenon; at the same time, delivering reliable forecasts is crucial to establish rational seismic risk reduction strategies, and to help communities prepared for potentially destructive earthquakes. This study derives the probability of extreme events in any seismic cluster generated by the Epidemic Type Aftershock Sequence (ETAS) model, a benchmark model in statistical seismology which belongs to the class of self-exciting, branching, Hawkes processes. This probability is obtained as a function of time, space and magnitude. The results we obtained contribute to understand the distinguishing features between mainshocks and foreshocks within a seismic sequence, and provide insights into earthquake prediction and probability assessment of extreme events in operational forecasting. Joint wirk with Petrillo Giuseppe (Institute of Statistical Mathematics, Tokyo) and Jiancang Zhuang (Institute of Statistical Mathematics, Tokyo).

### Scaling limits for parking on Frozen Erdős–Rényi Cayley trees with heavy tails

**Andrej Srakar** *University of Ljubljana*

In a recent contribution, Contat and Curien (2021; 2023) studied parking problem on uniform rooted Cayley tree with n vertices and m cars arriving sequentially, independently, and uniformly on its vertices. In a previous contribution, Lackner and Panholzer (2016) established a phase transition for this process when m ≈ n/2 . Contat and Curien couple this model with a variant of the classical Erdős–Rényi random graph process which enables describing the phase transition for the size of the components of parked cars using a »frozen« modification of the multiplicative coalescent, which freezes components with surplus. They showed scaling limit convergence in Skorokhod topology towards the frozen multiplicative coalescent and growth-fragmentation trees canonically associated to the 3/2-stable process (Bertoin, 2017). We study their model in the presence of group arrival of cars with power-law tail, and derive the appropriate metric space scaling limits in a Gromov-weak and Gromov-Hausdorff-Prokhorov topology using results on configuration model (Bhamidi et al., 2018; Broutin et al., 2020; Conchon-Kerjan and Goldschmidt, 2020; Dhara et al., 2020). We compare the results to more commonly studied Bienaymé–Galton–Watson trees, as well as study extensions to generalized frozen process (Contat and Curien, 2023) and analysis in the near supercritical regime.

### Gradient flow on control space with rough initial condition

**Florin Suciu ***Université Paris Dauphine-PSL*

We consider the (sub-Riemannian type) control problem of finding a path going from an initial point x to a target point y, by only moving in certain admissible directions. We assume that the corresponding vector fields satisfy the Hörmander condition, so that the classical Chow–Rashevskii theorem guarantees the existence of such a path. One natural way to try to solve this problem is via a gradient flow on control space. However, since the corresponding dynamics may have saddle points, any convergence result must rely on suitable (e.g. random) initialisation. We consider the case when this initialisation is irregular, which is conveniently formulated via rough path theory. In some simple cases, we manage to prove that the gradient flow converges to a solution, if the initial condition is the path of a Brownian motion (or rougher). The proof is based on combining ideas from Malliavin calculus with Łojasiewicz inequalities. A possible motivation for our study comes from the training of deep Residual Neural Nets, in the regime when the number of trainable parameters per layer is smaller than the dimension of the data vector. Joint work with Paul Gassiat (Paris Dauphine-PSL).

### Probabilistic interpretation of PDEs in a McKean-Vlasov case. An application to marble sulphation in Cultural Heritage

**Leonardo Tarquini** *Università degli Studi di Milano*

We discuss a probabilistic interpretation of a deterministic PDE-ODE reaction-diffusion system introduced for the description at the macroscale of the marble sulphation in Cultural Heritage (Natalini et al., 2004, 2007). This phenomenon occurs when sulphur dioxide reacts with the calcium carbonate rock, causing it to degrade. We derive a non-Markovian, McKean-type stochastic differential equation whose time marginal density satisfies a regularised version of the deterministic system. This is done with two different, although equivalent, approaches. The first one is based on analytical arguments, following a Feynman-Kač approach. The second one is obtained using probabilistic tools, such as diffusions with killing. Also, the second approach is very successful for numerical analysis. In both cases, we prove the well-posedness of the stochastic model, and we establish that propagation of chaos holds for the particle systems associated with such SDEs. Therefore, both stochastic systems can be interpreted as the dynamics of sulfur dioxide molecules during the reaction. In other words, a description of the degradation reaction at the microscale is obtained. This is a joint work with Daniela Morale and Stefania Ugolini (University of Milano).

### Stationary Mean-Field Games of Singular Control under Uncertainty

**Ioannis Tzouanas** *Bielefeld University*

In this (ongoing) work we study a class of stationary mean-field games (MFGs) of singular stochastic control under Knightian uncertainty. The representative agent adjusts the dynamics of an It\^o-diffusion via a two-sided singular stochastic control and faces a long-time-average expected profit criterion. The mean-field interaction is of scalar type and it is given through the stationary distribution of the population. Due to the presence of ambiguity, the problem of representative agent constructed as a stochastic game with two players. The decision maker chooses the 'best' policy and the adverse player the 'worst' probability measure. Via a constructive approach, we prove the existence and uniqueness of the stationary mean-field equilibrium. Furthermore, we show that this realizes a symmetric $\varepsilon_{N}$-Nash equilibrium for a suitable ergodic N-player game with singular controls. Finally, we provide explicit solution to an example of reversible investment MFG and we study the sensitivity of equilibria w.r.t. ambiguity. This is a joint work with Giorgio Ferrari.

### An analysis of Elo rating systems via Markov chains: bias, concentration, and tournament design

**Luca Zanetti** *University of Bath *

The Elo rating system is a popular method for calculating the relative skills of players (or teams) in sports analytics and particularly chess. It is based on a simple zero-sum update rule: if player A beats player B, then the rating of player A increases proportionally to the model probability that A would lose to B, while the rating of B decreases by the same amount. This amount depends on the previously calculated difference in skills between A and B. Despite their widespread popularity, Elo rating systems still lack a rigorous theoretical understanding (David Aldous called it “a neglected topic in applied probability”). Here, we take a probabilistic approach and study Elo under the well-known Bradley-Terry model. In this model, the Elo update can be seen as an iteration of stochastic gradient descent with fixed step-size for maximum likelihood estimation. Our approach, however, is to study Elo using tools from Markov chain theory. In particular, we study the mixing time of the underlying Markov chain, bound the inherent bias of the skills predicted by Elo wrt to the real skills of the players, and obtain concentration results about the time-averaged Elo ratings of the players. We also discuss the problem of tournament design: given a graph that represents the possible matchups between players, we want to construct a probability distribution over the edges of the graph that represents how often each matchup occurs. Our aim is to minimise the total number of matches that need to be played to approximate the real skills of the players. We show an intriguing connection between this problem and the fastest mixing Markov chain problem on a graph. Joint work with Sam Olesker-Taylor (University of Warwick).