Attractors for stochastic lattice dynamical systems with a multiplicative noise

In this paper, we consider a stochastic lattice differential equation with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and multiplicative white noise at each node. We prove the existence of a compact global random attractor which, pulled back, attracts tempered random bounded sets.      

On stationarity of stochastic retarded linear equations with unbounded drift operators

This article continues the study of Liu [Statist. Probab. Lett. 78(2008): 1775–1783; Stoch. Anal. Appl. 29(2011): 799–823] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a consequence, the associated stochastic equations have unbounded operators acting on the point or distributed delayed terms, while the operator acting on the instantaneous term generates a strongly continuous semigroup. We present conditions on the delay systems to obtain a unique stationary solution by combining spectrum analysis of unbounded operators and stochastic calculus. A few instructive cases are analyzed in detail to clarify the underlying complexity in the study of systems with unbounded delayed operators.     

Wiener Integrals with Respect to the Hermite Process and a Non-Central Limit Theorem

Abstract

We introduce Wiener integrals with respect to the Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an example, we study a generalization of the fractional Ornstein–Uhlenbeck process.     

A non-uniform discretization of stochastic heat equations with multiplicative noise on the unit sphere

We investigate a discretization of a class of stochastic heat equations on the unit sphere with multiplicative noise. A spectral method is used for the spatial discretization and the truncation of the Wiener process, while an implicit Euler scheme with non-uniform steps is used for the temporal discretization. Some numerical experiments inspired by Earth’s surface temperature data analysis GISTEMP provided by NASA are given.     

The effect of a generalized nonlinear incidence rate on the stochastic SIS epidemic model

In this paper, we aim to analyze the classical SIS epidemic model with a generalized force of infection (including nonmonotonic cases), where the transmission rate is perturbed by white noise. Using Feller's test for explosions, we prove that the disease dies out with probability one without any restriction on the model parameters.     

Weak martingale solutions to the stochastic Landau–Lifshitz–Gilbert equation with multi-dimensional noise via a convergent finite-element scheme

We propose an unconditionally convergent linear finite element scheme for the stochastic Landau–Lifshitz–Gilbert (LLG) equation with multi-dimensional noise. By using the Doss–Sussmann technique, we first transform the stochastic LLG equation into a partial differential equation that depends on the solution of the auxiliary equation for the diffusion part. The resulting equation has solutions absolutely continuous with respect to time. We then propose a convergent $\theta$-linear scheme for the numerical solution of the reformulated equation. As a consequence, we are able to show the existence of weak martingale solutions to the stochastic LLG equation.     

Rate of growth of the coalescent set in a coalescing stochastic flow

Consider an isotropic stochastic flow in R_d (i.e. a simultaneous random, correlated motion of all points in space), where d=l,2 or 3, such that the joint law of the motion of two particles allows the particles to meet and coalesce in finite time. The coalescent set J _t is a random subset of R_d consisting of the initial positions of particles which have coalesced by time t with the particle which started at 0. We show that the expected volume of J t grows at a rate proportional to when d=1, and at rates close to proportional to t/log t (resp. t) when d = 2 (resp. d=3). We give an example of a coalescing stochastic flow when d = 3. These results are analogous to growth rates of expected population size of a surviving type in the "invasion process" described by Clifford and Sudbury     

Analysis of a stochastic predator-prey system with foraging arena scheme

ABSTRACT

This paper focuses on a predator-prey system with foraging arena scheme incorporating stochastic noises. This SDE model is generated from a deterministic framework by the stochastic parameter perturbation. We then study how the correlations of the environmental noises affect the long-time behaviours of the SDE model. Later on the existence of a stationary distribution is pointed out under certain parametric restrictions. Numerical simulations are carried out to substantiate the analytical results.     

Strong convergence of a stochastic Rosenbrock-type scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretize the SPDE in space by the finite element method and propose a novel scheme called stochastic Rosenbrock-type scheme for temporal discretization. Our scheme is based on the local linearization of the semi-discrete problem obtained after space discretization and is more appropriate for such equations. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise and obtain optimal rates of convergence. Numerical experiments to sustain our theoretical results are provided.     

Asymptotic profiles of the endemic equilibrium of a diffusive SIS epidemic system with saturated incidence rate and spontaneous infection

A susceptible‐infected‐susceptible (SIS) epidemic reaction‐diffusion model with saturated incidence rate and spontaneous infection is considered. We establish the existence of endemic equilibrium by using a fixed‐point theorem. The global asymptotic stability of the constant endemic equilibrium is discussed in the case of homogeneous environment. We mainly investigate the effects of diffusion and saturation on asymptotic profiles of the endemic equilibrium. When the saturated incidence rate tends to infinity, the susceptible and infective distributes in the habitat in a nonhomogeneous way; this result is in strong contrast with the case of no spontaneous infection, where the endemic equilibrium tends to the disease free equilibrium. Our analysis shows that the spontaneous infection can enhance the persistence of an infectious disease and may provide some useful implications on disease control.     

Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise

The stochastic heat equation driven by additive noise is discretized in the spatial variables by a standard finite element method. The weak convergence of the approximate solution is investigated and the rate of weak convergence is found to be twice that of strong convergence. M. Kovács and S. Larsson supported by the Swedish Research Council (VR). Part of this work was done at Institut Mittag-Leffler. S. Larsson supported by the Swedish Foundation for Strategic Research (SSF) through GMMC, the Gothenburg Mathematical Modelling Centre.     

Stationary distribution and extinction of a stochastic one-prey two-predator model with Holling type II functional response

In this article, we investigate a stochastic one-prey two-predator model with Holling type II functional response. We first establish sufficient conditions for persistence and extinction of prey and predator populations, then by constructing a suitable stochastic Lyapunov function, we establish sharp sufficient criteria for the existence of a unique ergodic stationary distribution of the positive solutions to the model. The results show that the smaller white noise can ensure the persistence of prey and predator populations while the larger white noise can lead to the extinction of prey and predator populations.     

A note on the reflected backward stochastic differential equations driven by a Lévy process with stochastic Lipschitz condition

In this note, we prove the existence and uniqueness of the solution for a class of reflected backward stochastic differential equations (RBSDEs in short) related to the subdifferential operator of a lower semi-continuous convex function, driven by Teugels martingales associated with a Lévy process. Some known results are generalized and improved.     

A note on the stochastic properties of a scale change random effects model

The scale change model in survival analysis incorporates unobserved heterogeneity through a frailty that enters the baseline hazard function to change the time scale. In this paper we examine the stochastic properties of the mixtures of scale change model and build dependence between the overall population variable and the frailty variable. We also carry out stochastic comparisons between overall population variables when their respective frailty or baseline variables are ordered in the sense of various stochastic orders. Finally, we demonstrate how the variation of the baseline variable has an effect on the model.     

A class of stochastic processes with a law generalizing a nondecomposable law on the real line

In this paper we define a class of stochastic processes where law can be considered as a natural generalization of a nondecomposable law. In particular case, we express the processes thus defined as semimartingales with a Brownian martingale part, and compute the likelihood for detecting a signal immersed in additive noise which looks like Brownian motion, but has different independence properties.     

Asymptotics of the goodness-of-fit test for a partial linear model with randomly censored data

In this paper, we discuss the problem of testing the hypothesis that the underlying regression is a partial linear model. A test statistic, which is based on the quadratic form of a cusum process of residuals, is proposed. The asymptotic distributions of the test statistic under null hypothesis and the local alternative hypothesis are given. The number simulation shows that the test is available.