This survey develops a second-order viewpoint on point processes and on configuration-valued stochastic dynamics. We treat point processes as random counting measures on configuration space and emphasize the tools that govern first and second order: factorial moment measures, correlation functions, the pair-correlation function, Campbell-Mecke integrals, and conditional intensities. We then review algebraic classes with explicit correlation structure - determinantal, permanental/Cox, and Pfaffian point processes - highlighting how their kernels constrain repulsion or clustering through g(r). The second part turns to interacting particle systems and flows, using two-point functions to compare lattice models (exclusion, voter, contact), continuum birth-death and Glauber dynamics, and the associated BBGKY-type correlation hierarchies. A central case study is one-dimensional coalescing and annihilating systems: at fixed times they form Pfaffian point processes, yielding explicit formulas for ρ t (1) , ρ t (2) and short-range inhibition induced by collision history, and connecting to the Arratia flow. We conclude with open problems on Pfaffian models with controlled attraction, second-order classification of IPS, and multi-type extensions.
In this paper, we present a comprehensive analysis of the dynamics of a cubic operator defined by a non-stochastic matrix (containing at least one negative entry) on the one-dimensional simplex. This operator depends on two parameters. We identify all fixed points and some periodic points, and under certain conditions on the parameters, we describe the limit points. Furthermore, we determine a region in the parameter space where the corresponding operator exhibits chaotic behavior.
In this paper we propose a generalization of the basic partitioning algorithm proposed by T.J. Sheskin for computing steady state probability vector of a finite Markov chain. This algorithm generates an exact solution for steady state probabilities for any finite, irreducible Markov chain. Theoretically there is no imposed limit on the size of the Markov chain for which steady state probabilities can be obtained, but in practice, the method will produce round off errors. we propose to generalize the partitioning technique to cover all possible variants of the Sheskin algorithm. Our proposal, besides being a mathematical curiosity, it gives answer to several possible modifications (variations) suggested, by the author of this algorithm. In the numerical part we compared our generalization with standard CFTP algorithm.
The article is devoted to the study of the winding number of a Gaussian random field on the plane. It is known that the winding number of a non-degenerate random field can be presented as a difference of two point processes. The non-degeneracy is proved. It is further proved that the corresponding point processes are simple, locally finite, stationary and mixing.
We analyze the evolution of probability measures associated with stochastic differential equations with interaction (SDEWI) featuring constant diffusion coefficients. Our main results concern the propagation of the logarithmic Sobolev inequality (LSI); we prove that if the initial measure satisfies an LSI, the solution preserves this property almost surely. Additionally, we provide exact growth estimates for the Wasserstein-2 distance between two solutions driven by the same Brownian motion, improving upon general stability bounds by utilizing the constant diffusion structure. Finally, we derive bounds for the expected displacement of the measure from its initial state, characterizing the short-time behavior of the system.
This research investigated the approximate controllability of a class of second-order stochastic differential equations driven by Q-Wiener processes in a real separable Hilbert space. By employing techniques from stochastic analysis and functional analysis, we established the existence and uniqueness of mild solutions under relaxed conditions that are less stringent than the standard Lipschitz criterion. Furthermore, we proved the approximate controllability of the system using stopping time theory. Our findings contributed to the advancement of control theory for stochastic systems, particularly for second-order equations driven by Q-Wiener processes.
In this paper, we study the minimizers of U-processes and their domains of attraction. U-processes arise in various statistical contexts, particularly in M-estimation, where estimators are defined as minimizers of certain objective functions. Our main results establish necessary and sufficient conditions for the distributional convergence of these minimizers, identifying a broad class of normalizing sequences that go beyond the standard square-root asymptotics with normal limits. We show that the limit distribution belongs to exactly one of the four classes introduced by Smirnov. These results do not only extend Smirnov's theory but also generalize existing asymptotic theories for M-estimators, including classical results by Huber and extensions to higher-degree U-statistics. Furthermore, we analyze the domain of attraction for each class, providing alternative characterizations that determine which types of statistical estimators fall into a given asymptotic regime.
This paper investigates stochastic differential equations with interaction, introduced by Dorogovtsev the model of the evolution of large systems of interacting particles in random environments. The study emphasizes the difference approximation scheme for these equations, which involve approximating solutions in an infinite-dimensional, nonlinear space of measures. The key contributions include the formulation of approximation schemes for compactly supported initial measures, the derivation of Wasserstein distance-based estimates, and spatial discretization techniques.
The article is devoted to the asymptotic properties of Gaussian random field on the plane. We find the conditions for convergence of the number of upcrossings in the weak sense.
In this article we construct the random link with two components as the image of two non-intersecting circles in plane under smooth Gaussian random field. We study the properties of distributions of linking number and average crossing number of obtained link. We propose the results about existence of moments of linking number, average crossing number, Mobius energy of a link.
The classification of critical points of solutions to deterministic differential equation with interaction is proposed. The critical points of the type "A" and of the type "B" are introduced. The large time behavior of measure-valued solutions to such equations in the neighbourhoods of critical points of both types is studied.
In this paper we consider Potts-SOS model, with spin values 0, 1, 2, on the Cayley tree of order two. We study the weakly periodic Gibbs measures for this model, with respect to normal subgroup of two index of the group representation of a Cayley tree.
This paper explores the asymptotic behavior of solutions to multi-dimensional stochastic differential equations with interactions (SDEWI). By integrating interaction terms reflecting the distribution of all participating particles, SDEWI provides a complex model that captures dynamic changes across a system of particles. Theoretical insights are substantiated by the derivation of conditions under which the solutions exhibit specific asymptotic properties. This work extends previous research by confirming the shift-compactness criterion and establishing conditions for the existence of asymptotic limits, thereby offering a deeper understanding of the interaction dynamics within stochastic systems.
An arccos-quasi-quadratic metric is defined on a subset of R d+1 such as a sphere, a ball, an ellipsoidal surface, an ellipsoid, a simplex, a conic surface, or a hyperbolic surface, and the corresponding metric space incorporates several important cases in a unified framework that makes possible for us to study metric-dependent random fields on different metric spaces in a unified manner. Over the arccos-quasi-quadratic metric space, this paper constructs a class of time-varying vector random fields via either spherical harmonics or ultraspherical polynomials, and builds up various parametric and semiparametric covariance matrix structures. The extension problem is discussed as well.
We consider a variation of the secretary problem, in which n objects have random lifetimes after their scan. After scanning each object in turn, an observer either continues the process or chooses one of the scanned objects. In the latter case, the selection process ends, although the selected object may already disappear. We assume that its lifetime has a geometric distribution with parameter α. Unlike all previous formulations of the secretary problem, in which the scanning started with the first object, we investigate the optimal strategy of a belated observer who joined the selection process with a delay. In the final part of the article, we investigate the transient regime when α = α (n) → 1 as n → ∞. It turns out that the behavior of the threshold level L = L(n) can be arbitrary, as long as lim n → ∞ L/n ≥ 1/e.
In the article the averaged integral of the Lévy-driven linear process weighted by the complex exponential of a polynomial with real coefficients is considered. It is proved that uniformly over all real coefficients values of this polynomial such an averaged integral tends to zero a.s. It is also shown how the result obtained can be used to prove the LSE strong consistency of the chirp signal parameters.
The itinerary from the notion of a diffusion process to that of a generalized diffusion process is split into six lessons. Numerous exercises throughout its extent make this minicourse look like a collection of etudes for those ones who are interested in the theory of diffusion processes.
This paper considers a critical reduced process generated by a Galton-Watson branching process in which the number offspring of one particle possibly has infinite variance and conditional limit theorem proved. The rate of weak convergence of the critical reduced processes to the limit law is also obtained in the case when the number offspring of one particle has a finite variance.
A convex hull generated by the implementation of a Poisson point process inside a parabola is considered in the article. At that, the measure of intensity of the Poisson law is related to regularly varying functions near the boundary of the support. It is proven that the domain bounded by the perimeters of the convex hull and the boundary of the support - a parabola, can be represented as a sum of independent identically distributed random variables. Moreover, this value does not depend on the vertices of the convex hull itself. It is worth noting that having approximated the binomial point process by a Poisson one, P.Groeneboom [6], A.J.Cabo and P.Groeneboom [3], I.Hueter [9], T.Hsing [8] and others, using the martingale properties of stationary vertex processes, proved various options of the central limit theorem for functionals of a random convex hull in the case when the original distribution is uniformly concentrated in a convex polygon or ellipse. In this paper, the exact distribution and conditional distribution of vertex processes are found when the convex hull is generated by a inhomogeneous Poisson point process inside a parabola. In some special cases, it is shown that the area between the perimeter of the convex hull and the support of the distribution is expressed by the sum of independent random variables.
In this paper, we study properties of random walks on finite groups and later use them to obtain the limiting braid length expectation and component number of braid closure in a model of random braids, which is constructed by lifting elements of random walk on a Coxeter group to a braid group.
