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We study a compound Poisson (random time-change) approximation for stochastic differential equations (SDEs) and stochastic Volterra equations whose coefficients may be merely measurable in time and may even exhibit integrable singularities.…

Let $(X_1, \xi_1), (X_2,\xi_2),\ldots$ be i.i.d.~copies of a pair $(X,\xi)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\infty)$ and $\xi$ is a positive random variable. Define $S_k := \xi_1+\ldots+\xi_k$, $k \in…

We establish finite-dimensional central limit theorems for local, additive, interaction functions of temporally evolving point processes. The dynamics are those of a spatial Poisson process on the flat torus with points subject to a…

The Vlasov-Poisson system, modeling the evolution of non-collisional plasmas in the electrostatic limit, is approx- imated by a Semi-Lagrangian technique. Spectral methods of periodic type are implemented through a collocation approach.…

In this paper, we proved moderate deviation principles for a fully coupled two-time-scale stochastic systems, where the slow process is given by stochastic differential equations with small noise, while the fast process is a rapidly…

This work deals with the one-dimensional Stefan problem with a general time-dependent boundary condition at the fixed boundary. Stochastic solutions are obtained using discrete random walks, and the results are compared with analytic…

The standard small-time functional central limit theorem of semimartingales has been established in (Gerhold, S., Kleinert, M., Porkert, P., and Shkolnikov, M. (2015). Small time central limit theorems for semimartingales with applications.…

This exposition explains the basic ideas of Stein's method for Poisson random variable approximation and Poisson process approximation from the point of view of the immigration-death process and Palm theory. The latter approach also enables…

A second-order Galton-Watson process with immigration can be represented as a coordinate process of a 2-type Galton-Watson process with immigration. Sufficient conditions are derived on the offspring and immigration distributions of a…

The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson…

In this paper, we establish a local theory, i.e., existence, uniqueness and blow-up criterion, for a general family of singular SDEs in some Hilbert space. The key requirement is an approximation property that allows us to embed the…

We study a stochastic differential equation driven by a Poisson point process, which models continuous changes in a population's environment, as well as the stochastic fixation of beneficial mutations that might compensate for this change.…

In the paper the rescaled occupation time fluctuation process of a certain empirical system is investigated. The system consists of particles evolving independently according to \alpha-stable motion in R^d, \alpha<d<2\alpha. The particles…

In this paper, we address the full discretization of Friedrichs' systems with a two-field structure, such as Maxwell's equations or the acoustic wave equation in div-grad form, cf. [14]. We focus on a discontinuous Galerkin space…

Continuous-time stochastic processes play an important role in the description of random phenomena, it is therefore of prime interest to study particular variables depending on their paths, like stopping time for example. One approach…

Motivated by the recent contribution \cite{BB17} we study the scaling limit behavior of a class of one-dimensional stochastic differential equations which has a unique attracting point subject to a small additional repulsive perturbation.…

We investigate the relaxation problem and the diffusion phenomenon for the compressible Euler system with a time-dependent damping coefficient of the form $\tfrac{\mu}{(1+t)^{\lambda}}$ in $\mathbb{R}^d$ $(d \geq 1)$. We establish uniform…

We consider the random walk on a simple point process on $\Bbb{R}^d$, $d\geq2$, whose jump rates decay exponentially in the $\alpha$-power of jump length. The case $\alpha =1$ corresponds to the phonon-induced variable-range hopping in…

The continuous time Markov process considered in this paper belongs to a class of population models with linear growth and catastrophes. There, the catastrophes happen at the arrival times of a Poisson process, and at each catastrophe time,…

For a spectrally positive strictly stable process with index in (1,2), the paper obtains i) the density of the time when the process makes first exit from an interval by hitting the interval's lower end point before jumping over its upper…