Poster session 4
Thursday, 7/17, 12:00-13:00, Congress Centre
P43: PERIODIC SOLUTION OF A STOCHASTIC EPIDEMIC MODEL WITH TWO DIFFERENT EPIDEMICS AND DIFFERENT TRANSMISSION MECHANISM
Shivam Kumar Mishra (Indian Institute of Technology Mandi, India)
stand: S1
In epidemiology, the host population may be at risk for more than one infectious disease. Researchers have been recently interested in this area of study. In the article, we have explored an epidemic model that combines SIRS and SIR, two different transmission techniques. The considered deterministic model has been perturbed stochastically at transmission rates. For the resulting stochastic model, the analysis has been done. We study the periodic solution for the stochastic system. We use the Lyapunov function and Khasminskii theory to establish that the nonautonomous periodic form of the system with white noise has a positive periodic solution.
Bibliography
\([1]\) Cantab, M. D., Hamer, W. H. (2006). The Milroy lectures on epidemic disease in England—The evidence of variability and persistence of type. The Lancet 167(4305): 569–574.
\([2]\) Kermack, W. O., McKendrick, A. G. (1991). Contributions to the mathematical theory of epidemics—-I, Bulletin of Mathematical Biology 53(1-2): 33-55.
\([3]\) Jin, Z., Li, G. (2005). Global stability of an SEI epidemic model with general contact rate. Chaos, Solitons & Fractals 23(3): 997–1004.
\([4]\) Meng, X., Wu, Z., Zhang, T. (2013). The dynamics and therapeutic strategies of a SEIS epidemic model. International Journal of Biomathematics 6(5):, 1793-5245.
\([5]\) Bao, K., Rong, L., Zhang, Q. (2019). Analysis of a stochastic SIRS model with interval parameters. Discrete and Continuous Dynamical Systems-B 24(9), 4827- 4849.
\([6]\) Gray, A., Greenhalgh, D., Hu, L., Mao, X., Pan, J. (2011). A Stochastic Differential Equation SIS Epidemic Model. SIAM Journal on Applied Mathematics 71(3): 876-902.
\([7]\) Chang, Z., Lu, X., Meng, X. (2017). Analysis of a novel stochastic SIRS epidemic model with two different saturated incidence rates. Physica A: Statistical Mechanics and its Applications 472: 103-116.
\([8]\) Meng, X. (2010). Stability of a novel stochastic epidemic model with double epidemic hypothesis. Applied Mathematics and Computation 217(2): 506-515.
\([9]\) Feng, T., Meng, X., Zhang, T., Zhao, S. (2016) Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis. Journal of Mathematical Analysis and Applications 433(1): 227–242.
P44: Limit Theorems for the Infinite Occupancy Scheme
Valeriia Kotelnykova (University of California, Irvine; Taras Shevchenko National University of Kyiv)
stand: S2
The infinite occupancy scheme is defined as follows. Balls are thrown independently into an infinite array of boxes numbered \(1\), \(2,\ldots\), with a probability \(p_k\) of hitting the box \(k\). Here, \((p_k)_{k\in\mathbb{N}}\) is a discrete probability distribution with infinitely many \(p_k>0\).
We are interested in the number of occupied boxes, denoted by \(\mathcal{K}_n\), after \(n\) balls have been distributed. First, I will discuss the common assumptions imposed on the distribution \((p_k)_{k\in\mathbb{N}}\).
Then, I will provide an overview of the recent limit theorems for the number of occupied boxes \(\mathcal{K}_n\): a functional limit theorem \([2]\); laws of the iterated and single logarithm \([1]\).
Bibliography
\([1]\) D. Buraczewski, A. Iksanov and V. Kotelnikova, Laws of the iterated and single logarithm for sums of independent indicators, with applications to the Ginibre point process and Karlin’s occupancy scheme. Stochastic Process. Appl. 183 (2025), Paper No. 104597.
\([2]\) A. Iksanov, Z. Kabluchko and V. Kotelnikova, A functional limit theorem for nested Karlin’s occupancy scheme generated by discrete Weibull-like distributions. J. Math. Anal. Appl. 507 (2022), 125798.
P46: Modelling of natural catastrophe losses in adjacent regions
Martyna Zdeb (Politechnika Wrocławska)
stand: S4
Natural catastrophes may cause enormous losses in property and infrastructure, posing a huge threat not only to societies, but insurance and reinsurance companies as well. At the moment, catastrophe bonds are widely used to transfer risks related to natural catastrophes from companies to capital market investors. Pricing of a CAT bond involves modelling of the occurrence and the severity of losses caused by catastrophic events. Since CAT bonds are usually affecting multiple regions at the same time, we construct different pricing models based on various scenarios of dependence between losses in different areas. We consider cases with independent loss amounts, constant proportion between them, and arbitrary two-dimensional distributions with given correlation coefficient, and find differences between considered approaches. We illustrate considered models using data about catastrophe losses from Property Claim Services.
P47: Identification of the heavy-tailed behaviour using modified Greenwood statistic - univariate and multivariate case.
Katarzyna Skowronek (Politechnika Wrocławska)
stand: S5
In this work, we present a methodology based on a modified Greenwood
statistic for statistical testing in various scenarios for univariate
and multivariate data. Classical Greenwood statistic is defined for
positive random variables, while it’s modified version is defined for
all real-valued random variables. One of the most important properties
of the classical and modified Greenwood statistics is their stochastic
monotonicity within the star-shaped ordered distribution. We utilize
this property to propose statistical tests for the normal distribution
for the classes of \(\alpha\)-stable and Student’s t distributions, and
for infinite variance in the classes of Pareto and Student’s t
distributions.
Moreover, a relatively simple form of the modified Greenwood statistic
allows a straightforward extension to the multivariate case. We prove
that the stochastic monotonicity of the modified Greenwood statistic
holds for multivariate star-shaped ordered distributions. Therefore, we
expand the proposed tests from the univariate to the multivariate case.
In this work, we limit our analysis to the two- and three-dimensional
distributions.
To validate the efficiency of the proposed methods, we demonstrate the
results of the proposed methods via Monte Carlo simulations. In
addition, we compare introduced tests with the methods known in the
literature.
Bibliography
\([1]\) K. Skowronek, M. Arendarczyk, R. Zimroz, A. Wyłomańska: Modified Greenwood statistic and its application for statistical testing, Journal of Computational and Applied Mathematics 452, 116122, 2024
\([2]\) K. Skowronek, M. Arendarczyk, A. K. Panorska, T. J. Kozubowski, A. Wyłomańska: Testing and estimation of the index of stability of univariate and bivariate symmetric alpha-stable distributions via modified Greenwood statistic, Journal of Computational and Applied Mathematics 467, 116587, 2025
P48: Progressive intrinsic ultracontractivity and uniform ergodicity of discrete Feynman–Kac semigroups
Mateusz Śliwiński (Wrocław University of Science and Technology)
stand: S6
We present results of our investigation of a particular discrete-time counterpart of the Feynman–Kac semigroup with a confining potential in a countably infinite space. We focus on Markov chains with the direct step property, which is satisfied by a wide range of typically considered kernels. While intrinsic ultracontractivity is a very strong tool in studying the analytical and ergodic properties of these structures, in our setting it requires a sufficiently fast growth of the potential, severely restricting its applicability. In our joint work with Wojciech Cygan, René Schilling and Kamil Kaleta \([1,2]\), we use the concept of progressive intrinsic ultracontractivity (pIUC), originally introduced by Kaleta and Schilling in the continuous case, to prove that Feynman–Kac semigroups satisfying our assumptions, as well as their related intrinsic semigroups, enjoy a progressive analogue of uniform (quasi-)ergodicity.
Bibliography
\([1]\) Wojciech Cygan, Kamil Kaleta and Mateusz Śliwiński. "Decay of harmonic functions for discrete time Feynman-Kac operators with confining potentials" ALEA Lat. Am. J. Probab. Math. Stat., vol.19, no.1, 2022, pp.1071–1101.
\([2]\) Wojciech Cygan, Kamil Kaleta, René Schilling and Mateusz Śliwiński. "Heat kernels, intrinsic contractivity and ergodicity of discrete-time Markov chains killed by potentials" arXiv:2504.17879
P49: Convergence rate of Euler-Maruyama scheme for McKean-Vlasov SDEs with density-dependent drift
Anh-Dung Le (École nationale des ponts et chaussées)
stand: S7
In this paper, we study weak well-posedness of a McKean-Vlasov stochastic differential equations (SDEs) whose drift is density-dependent and whose diffusion is constant. The existence part is due to Hölder stability estimates of the associated Euler-Maruyama scheme. The uniqueness part is due to that of the associated Fokker-Planck equation. We also obtain convergence rate in weighted \(L^1\) norm for the Euler-Maruyama scheme.
P50: Preventing large-scale avalanches in the 2D Abelian sandpile model using strategic interventions
Maike de Jongh (University of Twente)
stand: S8
The Abelian sandpile model (ASM) is a fundamental example of a dynamical system exhibiting self-organized criticality, offering key insights into the emergence of extreme events. Avalanche dynamics in the ASM have inspired models of earthquakes, forest fires, solar flares, and cascading failures in networks. In our work, we explore how strategic interventions can reduce the expected size of avalanches.
First, we present a method to compute the expected size of an avalanche generated by a cluster of critical vertices, based on the decomposition of avalanches into a sequence of waves proposed by Ivashkevich et al. \([1]\). Second, we analyze the effect of removing sand grains from square-shaped clusters of critical vertices. For such clusters, we provide explicit results on the impact of targeted sand grain removals and identify the set of vertices from which removing sand grains most effectively reduces the expected avalanche size.
Bibliography
\([1]\) Ivashkevich, E.V., Ktitarev, D.V., Priezzhev, V.B. “Waves of topplings in an Abelian sandpile." Physica A, vol. 209, no. 3-4, 1994, pp. 347-360.
P51: Stochastic Dynamic Machine Scheduling with Interruptible Set-up Times
Dongnuan Tian (Lancaster University)
stand: S9
We consider a problem in which a machine is scheduled dynamically within a network to process jobs at different demand points. Costs are accumulated as jobs wait to be processed. The time needed for a machine to travel between two nodes in the network represents the “set-up time” needed to switch from processing one type of job to another. By using a network formulation, we can model complex relationships in switching times between different types of activity and also allow the switching times to be interrupted. The problem can be formulated as a Markov decision process in which arrival times, service times and switching times are uncertain and the objective is to minimize the expected long-run average cost. However exact solutions using dynamic programming are not possible due to the complexity of the state space. Heuristics for certain special cases of the problem have been proposed in the literature and in this talk we discuss how to adapt these heuristics to our problem. We also discuss how forward-thinking strategies can be used to develop improved heuristics and show the results obtained by applying these methods to networks of various different configurations.
P52: Windings of planar Stochastic Processes
Stavros Vakeroudis (Athens University of Economics and Business)
stand: S10
Two-dimensional (planar) processes attract the interest of several researchers. This happens both because of their richness from a theoretical point of view and because their study turns out to be very fruitful in terms of applications (e.g. in Finance \([9]\), in Biology \([8]\) etc.). Here, we focus on the fine study of trajectories of planar processes, and in particular on their windings.
We survey several results concerning windings of two-dimensional processes, including planar Brownian motion (BM), complex-valued Ornstein-Uhlenbeck (OU) processes and planar stable processes (see e.g. \([5,6]\). We also present Spitzer’s asymptotic Theorem for each case.
Our starting point is the skew-product representation. Then, we introduce Bougerol’s celebrated identity in law \([7]\) which is very useful for the study of the windings of planar BM and of complex-valued OU processes, stating that, for \(u>0\) fixed, \[\sinh(\beta_{u}) \stackrel{(law)}{=} \hat{\beta}_{A_{u}(\beta)=\int^{u}_{0}ds\exp(2\beta_{s})} \ ,\] where \((\beta_{t},t\geq0)\) and \((\hat{\beta_{t}},t\geq0)\) are two independent linear Brownian motions, and the second one is also independent from \(A_{u}(\beta)\). However, this method cannot be applied for the case of planar stable processes \([1]\). So, we tackle this problem firstly by using new methods invoking the continuity of the composition function \([3]\) and secondly by applying new techniques from the theory of self-similar Markov processes \([4]\) having as a starting point the so-called Riesz–Bogdan–Żak transform introduced in \([2]\) which gives the law of the stable process when passed through the spatial Kelvin transform and an additional time change. This approach allows to study similarly one-dimensional and (possibly) higher-dimensional windings.
Bibliography
\([1]\) Jean Bertoin and Wendelin Werner, Stable windings. Ann. Probab. 24(3), 1996, pp. 1269–1279.
\([2]\) Krzysztof Bogdan and Tomasz Żak, On Kelvin Transformation. J. Theor. Probab. 19, 2010, pp. 89–120.
\([3]\) Ron A. Doney and Stavros Vakeroudis, Windings of planar stable processes. Sém. Prob., Vol. XLV, Lecture Notes in Mathematics 2078, 2013, pp. 277–300.
\([4]\) Andreas E. Kyprianou and Stavros Vakeroudis, Stable windings at the Origin, Stoch. Proc. Appl. 128, 2018, pp. 4309-4325.
\([5]\) Stavros Vakeroudis, On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol’s identity. SIAM Theory Probab. Appl. 56(3), 2011, pp. 485–507 (originally published in 2011 in Teor. Veroyatnost. i Primenen., 56(3), pp. 566–591).
\([6]\) Stavros Vakeroudis, Bougerol’s identity in law and extensions. Probability Surveys 9, 2012, pp. 411–437.
\([7]\) Stavros Vakeroudis and Marc Yor, Integrability properties and Limit Theorems for the exit time from a cone of planar Brownian motion. Bernoulli 19(5A), 2012, pp. 2000–2009.
\([8]\) Stavros Vakeroudis, Marc Yor and David Holcman, The Mean First Rotation Time of a planar polymer. J. Stat. Phys. 143(6), 2011, pp. 1074–1095.
\([9]\) Marc Yor, Exponential Functionals of Brownian Motion and Related Processes. Berlin, Springer, 2001.
P53: Infinite-dimensional stochastic differential equations for Coulomb random point fields
Shota Osada (Kagoshima University)
stand: S11
The Coulomb random point field \(\mu\) is an infinite particle system in \(\mathbb{R}^d\) , \(d \geq 2\), defined by a sub-sequential limit of \[\notag \mu^N(d\xi) = \dfrac{1}{Z^{N}}\exp\left(-\beta \left\{\sum_{i=1}^{N}\Phi^N(x^i) + \sum_{j \neq k,\\j,k=1}^{N}\Psi(x^j-x^i)\right\}\right)d\mathbf{x}^N ,\] where \(\xi = \sum_{i=1}^{N}\delta_{x^i}\), \(\beta \geq 0\) is the inverse temperature, \(\{\Phi^N\}_{N \in \mathbb{N}}\) is a sequence of confining potentials, \(\Psi\) is the \(d\)-dimensional Coulomb potential defined by \[\Psi(x) = \dfrac{1}{d-2}\dfrac{1}{|x|^{d-2}} \,\,(d\geq 3) , \quad \Psi(x) =-\log{|x|} \,\, ( d=2).\]
This poster presents the existence of
\((\mathbb{R}^d)^{\mathbb{N}}\)-valued diffusion process
\(\mathbf{X}=(X^i)_{i \in \mathbb{N}}\) associated with the Coulomb random
point field for each \(d\geq 2\) and \(\beta \geq 0\), constructed as the
pathwise unique strong solution to the following infinite-dimensional
stochastic differential equation (ISDE): \[\tag{1}
X_t^i - X_0^i = \int_{0}^{t}\sigma(X_u^i)dB_u^i + \frac{\beta}{2}\int_{0}^{t}\nabla\mathfrak{a}(X_u^i)du\\
\quad -\dfrac{\beta}{2}\int_{0}^{t}\mathfrak{a}(X_u^i)\left(\nabla\Phi(X_u^i)
+\lim_{R \to \infty}\sum_{|X_u^i-X_u^j|\leq R , j \neq i}
\nabla\Psi(X_u^i-X_u^j)\right)du (i \in \mathbb{N}).\] Here,
\(\mathfrak{a} \in C_b^2(\mathbb{R}^d ; \mathbb{R}^{d\times d})\) is a
uniformly elliptic, bounded, symmetric matrix-valued function, and
\(\sigma \in C_b^{1}(\mathbb{R}^d; \mathbb{R}^{d\times d})\) is a
symmetric matrix such that \(\sigma^{t}\sigma = \mathfrak{a}\).
Let
\(\ell: \{\xi \in \mathrm{Conf}(\mathbb{R}^d)|\xi(\mathbb{R}^d)=\infty\} \rightarrow (\mathbb{R}^d)^{\mathbb{N}}\)
is a labeling map such that
\(\xi = \sum_{i \in \mathbb{N}}\delta_{\ell(\xi)^i}.\)
Theorem. Let \(\mu\) be a sub-sequential limit \(\mu = \lim_{n \to \infty} \mu^{N_n}\) such that
\(\sup_{n \in \mathbb{N}}\mathbf{E}^{\mu^{N_n}}[\xi(S_R)]<\infty\) for each \(R\in \mathbb{N}\).
\(\mu(\{\xi(\mathbb{R}^d)=\infty\})=1\)
\(\mathbf{E}^{\mu}[\xi(S_R)]=O(R^p)\) as \(R \to \infty\) for some \(p >0\) .
\(\mathbf{E}^{\mu} \left[\dfrac{\xi(S_R)!}{(\xi(S_R)-m)!}\right]<\infty\) for all \(m,R \in \mathbb{N}\).
Then, ISDE \((1)\) has a unique strong solution \(\mathbf{X}=(X^i)_{i \in \mathbb{N}}\) with starting point \(\mathbf{X}_0 = \ell(\xi)\) for \(\mu\)-a.s. \(\xi\).
P55: Stochastic Simulation for Transient Dynamics of Schrödinger’s Cat States
Yi Shi (University College London)
stand: S13
Schrödinger’s Cat states offer a valuable noise bias, with the potential to substantially simplify quantum error correction. However, simulating their transient dynamics is computationally demanding due to the high-dimensional, non-linear nature of these systems. This challenge is particularly acute when multiple interacting subsystems are involved, as the matrix dimension of the Lindblad master equation then expands exponentially. We address this by employing stochastic methods that utilize only a few stochastic variables per mode to accurately capture key observables in the transient dynamics.
P56: Littlewood–Paley estimates for pure-jump Dirichlet forms
Michał Gutowski (Wrocław University of Science and Technology)
stand: S14
Littlewood–Paley square functions were first introduced by Littlewood and Paley in \([1]\). They found application in many research areas, for instance, in harmonic analysis, in the study of \(L^p\)-spaces, Fourier multipliers, and partial differential equations.
We used a recently proved generalized Hardy–Stein identity to extend previous Littlewood–Paley estimates to pure-jump Dirichlet forms. Our results generalize earlier estimates obtained for pure-jump Lévy processes on Euclidean space \([2]\). Moreover, we relax some of the assumptions used in previous papers \([2, 3]\). To overcome the difficulty that Itô’s formula is not applicable in our setting, we employed the theory of Revuz correspondence and additive functionals. Meanwhile, we also present some counterexamples demonstrating that certain inequalities do not hold in the generality considered in our work.
Bibliography
\([1]\) John E. Littlewood, Raymond E. A. C. Paley. "Theorems on Fourier series and power series (II)." Proc. London Math. Soc. (2), vol. 42, no. 1, 1936, pp. 52–89.
\([2]\) Rodrigo Bañuelos, Krzysztof Bogdan, Tomasz Luks. "Hardy–Stein identities and square functions for semigroups." J. Lond. Math. Soc. (2), vol. 94, no. 2, 2016, pp. 462–478.
\([3]\) Huaiqian Li, Jian Wang. "Littlewood-Paley-Stein estimates for non-local Dirichlet forms." J. Anal. Math., vol. 143, no. 2, 2021, pp. 401–434.