This study investigates the weak convergence of Markov processes on discrete state spaces under the assumption that the transition intensities converge to a constant. Additionally, the research determines the limits of higher-order moments of the Markov process, which are utilized to prove the existence of limit theorems.

Let $(X_t)_{t \ge 0}$ be a diffusion process defined on a compact connected Riemannian manifold $M$, and for $\alpha > 0$, let $$ \mu_t^{(\alpha)} = \frac{\alpha}{t^\alpha} \int_{0}^{t} \delta_{X_s} \, s^{\alpha - 1} \mathrm{d} s $$ be the associated weighted empirical measure. We investigate asymptotic behavior of $\mathbb{E}^\nu \big[ \mathrm{W}_2^2(\mu_t^{(\alpha)}, \mu) \big]$ for sufficient large $t$, where $\mathrm{W}_2$ is the quadratic Wasserstein distance, $\nu$ is the law of $X_0$ and $\mu$ is the invariant measure of the process. In the particular case $\alpha = 1$, our result sharpens the limit theorem achieved in \cite{WZ19}. The proof is based on the PDE and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan.

The purpose of this paper is to investigate the existence of the Onsager-Machlup action functional for degenerate McKean-Vlasov stochastic differential equations. To this end, we first derive Onsager-Machlup action functional for degenerate McKean-Vlasov stochastic differential equations with constant diffusion in a broad set of norms by Girsanov transformation, some conditioned exponential inequalities and It $\mathrm{\hat{o}}$ formulas for distribution dependent functional. Then an example is given to illustrate our results.

We study a random tree, which was introduced in [4] as part of a model of a neuronal network. Realising a scaling relation for the law of the tree, we can use elementary techniques to derive asymptotic results on the geometry as time goes to infinity.

The relativistic equations of gravitation and electromagnetism in the form of Vlasov – Einstein – Maxwell equations are proposed and analyzed. For weakly relativistic equations we get an analog of Mealn – McCree solution. We also study Lagrange points in non-relativistic case with Einstein lambda- term.

This article provides information on Hermite polynomials and its application to some problems in risk theory and site percolation.

We prove a uniform exponential localization of eigenfunctions and simplicity of spectrum for a class of limit-periodic lattice Schr¨odinger operators. An important ingredient of the proof is a generalized variant of the well-known Minami estimates (correlation inequalities for the eigenvalues) to the case where the spectral intervals can be arbitrarily placed in the real line. The new corre- lation inequalities allow us to substantially simplify and make more transparent the application of the KAM (Kolmogorov-Arnold-Moser) techniques.

This survey concerns probabilistic models motivated by cooperative sequential adsorption (CSA) models. CSA models are widely used in physics and chemistry for modelling adsorption processes in which adsorption rates depend on the spatial configuration of already adsorbed particles. Corresponding probabilistic models describe random sequential allocation of particles either in a subset of Euclidean space, or at vertices of a graph (e.g. sites of the lattice). Depending on a technical setup these probabilistic models are stated in terms of spatial or integer-valued interacting birth-and-death processes. In this survey we consider several such models that have been studied in recent years.

The aim of this paper is twofold: - To give an elementary and self-contained proof of an explicit formula for the free energy for a general class of polymer chains interacting with an environment through periodic potentials. This generalizes a result in (6) in which the formula is derived by using the Donsker-Varadhan Large Deviations theory for Markov chains. We exploit instead tools from renewal theory. - To identify the infinite volume limits of the system. In particular, in the different regimes we encounter transient, null recurrent and positive recurrent processes (which correspond to delocalized, critical and localized behaviors of the trajectories). This is done by exploiting the sharp estimates on the partition function of the system obtained by the renewal theory approach. The precise characterization of the infinite volume limits of the system exposes a non-uniqueness problem. We will however explain in detail how this (at first) surprising phenomenon is instead due to the presence of a first-order phase transition. 2000 Mathematics Subject Classification: 60K35, 82B41, 82B44