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In this paper, a stochastic SIS epidemic infectious diseases model with double stochastic perturbations is proposed. First, the existence and uniqueness of the positive global solution of the model are proved. Second, the controlling conditions for the extinction and persistence of the disease are obtained. Besides, the effects of the intensity of volatility ξ 1 and the speed of reversion 1 on the dynamical behaviors of the model are discussed. Finally, some numerical examples are given to support the theoretical results. The results show that if the basic reproduction number ℛ 0 s < 1 , the disease will be extinct, that is to say that we can control the threshold ℛ 0 s to suppress the disease outbreak. [ABSTRACT FROM AUTHOR]

This paper is concerned with the attracting set of an ant colonies model with bounded noisy perturbation. This perturbation is modeled by the well-known Ornstein–Uhlenbeck process and the arc tangent function. For the random model, we first verify the existence and uniqueness of the global positive solution, and then prove the existence of the deterministic attracting set. Furthermore, in order to reveal more detailed information about the long-time behavior of the ant colonies system, we analyze the internal structure of the attracting set and provide some conditions under which coexistence (or extinction) of the ant species exists in the ant colonies system. [ABSTRACT FROM AUTHOR]

According to the transmission characteristics of COVID-19, this paper proposes a stochastic SAIRS epidemic model with two mean reversion Ornstein–Uhlenbeck processes and saturated incidence rates. We first prove the existence and uniqueness of the global solution in the stochastic model. Using several suitable Lyapunov methods, we then derive the extinction and persistence of COVID-19 under certain conditions. Further, stationary distribution and ergodic properties are obtained. Moreover, we obtain the probability density function of the stochastic model around the equilibrium. Numerical simulations illustrate our theoretical results and the effect of essential parameters. Finally, we apply the model to investigate the latest outbreak of the COVID-19 epidemic in Guangzhou city, China. [ABSTRACT FROM AUTHOR]

In this paper, we study a smooth approximation of an arbitrary càdlàg Lévy process. Such approximation processes are known as integrated fast oscillating Ornstein–Uhlenbeck (OU) processes. We know that approximating processes are continuous, while the limit of processes may be discontinuous, so convergence in uniform topology or Skorokhod J 1 -topology will not hold in general. Therefore, we establish an approximation in Skorokhod M 1 -topology. Note that the convergence is almost surely, which is an extension result of Hintze and Pavlyukevich. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Ornstein–Uhlenbeck (OU) process defined as solution to the equation d X t = − X t d t + d G t , X 0 = 0 , where { G t , t ≥ 0 } is a Gaussian process with stationary increments, whereas > 0 is unknown parameter to be estimated. We provide an upper bound in Kolmogorov distance for normal approximation of the least squares estimator ̃ T of the drift parameter on the basis of the continuous observation { X t , t ∈ [ 0 , T ] } , as T → ∞. Our approach is based on some novel estimates involving a combination of Malliavin calculus and Stein's method for normal approximation. We apply our result to fractional OU processes of the first kind, and improved the upper bound of the Kolmogorov distance for the LSE ̃ T provided by [Y. Chen, N. Kuang and Y. Li, Berry–Esseen bound for the parameter estimation of fractional Ornstein–Uhlenbeck processes, Stoch. Dyn. 20(4) (2020) 2050023; Y. Chen and Y. Li, Berry–Esseen bound for the parameter estimation of fractional Ornstein–Uhlenbeck processes with the hurst parameter H ∈ (0 , 1 2) , Commun. Stat. Theory Methods 50(13) (2021) 2996–3013], respectively, in the cases H ∈ (0 , 1 2) and H ∈ [ 1 2 , 3 4). We also apply our approach to fractional OU processes of the second kind. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of maximum likelihood estimators for Ornstein–Uhlenbeck process with large linear drift d X t = − 1 ( X t − 1 2 ν) d t + d B t , 0 ≤ t ≤ T , where , ν ∈ ℝ , and { B t } t ≥ 0 is a given standard Brownian motion. The law of iterated logarithm, consistency and asymptotic distributions of the estimators are discussed based on the continuous observation { X t } t ∈ [ 0 , T ] as → 0. [ABSTRACT FROM AUTHOR]

We study stochastic partial differential equations (SPDEs) with potentially very rough fractional noise with Hurst parameter H ∈ (0 , 1). Close to a change of stability measured with a small parameter , we rely on the natural separation of time-scales and establish a simplified description of the essential dynamics. Up to an error term bounded by a power of depending on the Hurst parameter we can approximate the solution of the SPDE in first order by an SDE, the so-called amplitude equation, which describes the amplitude of the dominating pattern changing stability. In second order the approximation is given by a fast infinite-dimensional Ornstein–Uhlenbeck process. To this aim, we need to establish an explicit averaging result for stochastic integrals driven by rough fractional noise for small Hurst parameters. [ABSTRACT FROM AUTHOR]

A particle roaming in a potential in the presence of spatial heterogeneities such as traps or physical barriers is a mathematical representation of a vast number of processes across multiple disciplines, ranging from the dynamics of chemical reactions and the movement of foraging animals to the performance of a financial asset. Among the approaches to tackle such problems, a powerful one is the so-called defect technique. Within a continuous space–time formulation, the technique has been successfully used to analyze diffusion problems with partially absorbing boundaries, but has not been employed to study other types of boundaries. Here we exploit the continuity equation to extend the defect technique to when partially reflecting barriers are present. For each boundary type, we apply the defect formalism to the Ornstein–Uhlenbeck process, recovering known analytical results and presenting new ones. [ABSTRACT FROM AUTHOR]

We aim to obtain a homogenization theorem for a passive tracer interacting with a fractional, possibly non-Gaussian, noise. To do so, we analyze limit theorems for normalized functionals of Hermite–Volterra processes and extend existing results to cover power series with fast decaying coefficients. We obtain either convergence to a Wiener process, in the short-range dependent case, or to a Hermite process, in the long-range dependent case. Furthermore, we prove convergence in the multivariate case with both, short- and long-range dependent components. Applying this theorem, we obtain a homogenization result for a slow/fast system driven by such Hermite noises. [ABSTRACT FROM AUTHOR]

In this paper, we consider the maximum likelihood estimation for the symmetric α -stable Ornstein–Uhlenbeck (S α S-OU) processes based on discrete observations. Since the closed-form expression of maximum likelihood function is hard to obtain in the Lévy case, we choose a mixture of Cauchy and Gaussian distribution to approximate the probability density function (PDF) of the S α S distribution. By means of transition function and Laplace transform, we construct an explicit approximate sequence of likelihood function, which converges to the likelihood function of S α S distribution. Based on the approximation of likelihood function we give an algorithm for computing maximum likelihood estimation. We also numerically simulate some experiments which demonstrate the accuracy and stability of the proposed estimator. [ABSTRACT FROM AUTHOR]

We consider the infinite dimensional Heston stochastic volatility model proposed in Ref. 7. The price of a forward contract on a non-storable commodity is modeled by a generalized Ornstein–Uhlenbeck process in the Filipović space with this volatility. We prove a representation formula for the forward price. Then we consider prices of options written on these forward contracts and we study sensitivity analysis with computation of the Greeks with respect to different parameters in the model. Since these parameters are infinite dimensional, we need to reinterpret the meaning of the Greeks. For this we use infinite dimensional Malliavin calculus and a randomization technique. [ABSTRACT FROM AUTHOR]

This paper is about the structure of all entrance laws (in the sense of Dynkin) for time-inhomogeneous Ornstein–Uhlenbeck processes with Lévy noise in Hilbert state spaces. We identify the extremal entrance laws with finite weak first moments through an explicit formula for their Fourier transforms, generalizing corresponding results by Dynkin for Wiener noise and nuclear state spaces. We then prove that an arbitrary entrance law with finite weak first moments can be uniquely represented as an integral over extremals. It is proved that this can be derived from Dynkin's seminal work "Sufficient statistics and extreme points" in Ann. Probab. 1978, which contains a purely measure theoretic generalization of the classical analytic Krein–Milman and Choquet Theorems. As an application, we obtain an easy uniqueness proof for T -periodic entrance laws in the general periodic case. A number of further applications to concrete cases are presented. [ABSTRACT FROM AUTHOR]

For the Ornstein–Uhlenbeck process in stationary and explosive cases, this paper studies Cramér-type moderate deviations for the log-likelihood ratio process. As an application, we give the negative regions of drift testing problem, and also obtain the decay rates of the error probabilities. The main methods of this paper consist of mod- ϕ convergence approach, deviation inequalities for multiple Wiener–Itô integrals and asymptotic analysis techniques. [ABSTRACT FROM AUTHOR]

A generalized competitive system with stochastic perturbations is proposed in this paper, in which the stochastic disturbances are described by the famous Ornstein–Uhlenbeck process. By theories of stochastic differential equations, such as comparison theorem, Itô's integration formula, Chebyshev's inequality, martingale's properties, etc., the existence and the uniqueness of global positive solution of the system are obtained. Then sufficient conditions for the extinction of the species almost surely, persistence in the mean and the stochastic permanence for the system are derived, respectively. Finally, by a series of numerical examples, the feasibility and correctness of the theoretical analysis results are verified intuitively. Moreover, the effects of the intensity of the stochastic perturbations and the speed of the reverse in the Ornstein–Uhlenbeck process to the dynamical behavior of the system are also discussed. [ABSTRACT FROM AUTHOR]

This paper considers a class of Levy process namely the variance gamma (VG) process to offer a more realistic way to model the dynamics of the logarithm of stock prices. Then, we verify the uniqueness and existence of the solution to the stochastic differential equation of the model. We also examine the valuation of multi-asset American options under VG model when the correlation coefficient is governed by the modified Ornstein–Uhlenbeck process. Various simulation experiments are presented and the achieved results are tested empirically for option prices using S&P 500 data. [ABSTRACT FROM AUTHOR]

In this paper, we present a detailed study of the electronic dynamics in systems with correlated disorder generated from the Ornstein–Uhlenbeck process (OU). In short, we used numeric methods for solving the time-dependent Schrödinger equation. We apply a Taylor's expansion of the evolution operator in order to solve the differential equation. We calculate some typical tools, such as the participation function ξ (t) , the mean square displacement σ (t) and the probability of return R (t). In our analysis, we show that for low correlations the system behaves as in the standard Anderson model (i.e. all eigenstates are localized). For strong correlations, our results suggest the existence of a quasi-ballistic dynamics. [ABSTRACT FROM AUTHOR]

For an Ornstein–Uhlenbeck process driven by fractional Brownian motion with Hurst index H ∈ [ 1 2 , 3 4 ] , we show the Berry–Esséen bound of the least squares estimator of the drift parameter based on the continuous-time observation. We use an approach based on Malliavin calculus given by Kim and Park [Optimal Berry–Esséen bound for statistical estimations and its application to SPDE, J. Multivariate Anal. 155 (2017) 284–304]. [ABSTRACT FROM AUTHOR]

We consider a least square-type method to estimate the drift parameters for the mean-reverting Ornstein–Uhlenbeck process of the second kind { X t , t ≥ 0 } defined as d X t = 𝜃 (μ + X t) d t + d Y t , G (1) , t ≥ 0 , with unknown parameters 𝜃 > 0 and μ ∈ ℝ , where Y t , G (1) : = ∫ 0 t e − s d G a s with a t = He t H , and { G t , t ≥ 0 } is a Gaussian process. In order to establish the consistency and the asymptotic distribution of least square-type estimators of 𝜃 and μ based on the continuous-time observations { X t , t ∈ [ 0 , T ] } as T → ∞ , we impose some technical conditions on the process G , which are satisfied, for instance, if G is a fractional Brownian motion with Hurst parameter H ∈ (0 , 1) , G is a subfractional Brownian motion with Hurst parameter H ∈ (0 , 1) or G is a bifractional Brownian motion with Hurst parameters (H , K) ∈ (0 , 1) × (0 , 1 ]. Our method is based on pathwise properties of { X t , t ≥ 0 } and { Y t , G (1) , t ≥ 0 } proved in the sequel. [ABSTRACT FROM AUTHOR]

We studied the existence of a random exponential attractor in the weighted space of infinite sequences for second-order nonautonomous stochastic lattice system with linear multiplicative white noise. Firstly, we present some sufficient conditions for the existence of a random exponential attractor for a continuous cocycle defined on a weighted space of infinite sequences. Secondly, we transferred the second-order stochastic lattice system with multiplicative white noise into a random lattice system without noise through the Ornstein–Uhlenbeck process, whose solutions generate a continuous cocycle on a weighted space of infinite sequences. Thirdly, we estimated the bound and tail of solutions for the random system. Fourthly, we verified the Lipschitz continuity of the continuous cocycle and decomposed the difference between two solutions into a sum of two parts, and carefully estimated the bound of the norm of each part and the expectations of some random variables. Finally, we obtained the existence of a random exponential attractor for the considered system. [ABSTRACT FROM AUTHOR]

In this paper, we consider the self-normalized asymptotic properties of the parameter estimators in the fractional Ornstein–Uhlenbeck process. The deviation inequalities, Cramér-type moderate deviations and Berry–Esseen bounds are obtained. The main methods include the deviation inequalities and moderate deviations for multiple Wiener–Itô integrals [P. Major, Tail behavior of multiple integrals and U-statistics, Probab. Surv. 2 (2005) 448–505; On a multivariate version of Bernsteins inequality, Electron. J. Probab. 12 (2007) 966–988; M. Schulte and C. Thäle, Cumulants on Wiener chaos: Moderate deviations and the fourth moment theorem, J. Funct. Anal. 270 (2016) 2223–2248], as well as the Delta methods in large deviations [F. Q. Gao and X. Q. Zhao, Delta method in large deviations and moderate deviations for estimators, Ann. Statist. 39 (2011) 1211–1240]. For applications, we propose two test statistics which can be used to construct confidence intervals and rejection regions in the hypothesis testing for the drift coefficient. It is shown that the Type II errors tend to be zero exponential when using the proposed test statistics. [ABSTRACT FROM AUTHOR]

The dynamics of the tripartite thermal entanglement measured by Negativity (N) and the tripartite quantum correlation described by measurement-induced disturbance (MID) under Ornstein–Uhlenbeck noise are investigated. This study has found that the tripartite N and MID can be preserved more effectively in the non-Markovian environment than in the short-time limit and the Markov limit cases. The short-time limit is a better approximation than the Markov limit. MID vanishes only in the asymptotic limit, while entanglement sudden death may occur, and the decreasing duration of MID far outweighs entanglement. This implies that MID is more robust than Negativity. As the noise bandwidth increases, the disentanglement time and the decay time of MID are significantly shorter. The increase of XZX + YZY three-site interaction is more effective than XZY−YZX three-site interaction to enhance Negativity and MID as well as the disentanglement time. The magnetic field diminishes Negativity and MID, but has no significant influence on the decreasing durations of both Negativity and MID. [ABSTRACT FROM AUTHOR]

This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein-Uhlenbeck semigroup . Our approach is to expand the Mehler kernel into Hermite polynomials and apply the powers of the Ornstein-Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for . As an application we give an alternative proof of the kernel estimates by Ref. 10, making all relevant quantities explicit. [ABSTRACT FROM AUTHOR]

We study the effect of additive colored noise on the evolution of maps and demonstrate that the deviations caused by such noise can be reduced using coupled dynamics. We consider both Ornstein-Uhlenbeck process as well as noise in our numerical simulations. We observe that though the variance of deviations caused by noise depends on the correlations in the noise, under optimal coupling strength, it decreases by a factor equal to the number of coupled elements in the array as compared to the variance of deviations in a single isolated map. This reduction in noise levels occurs in chaotic as well as periodic regime of the maps. Lastly, we examine the effect of colored noise in chaos computing and find that coupling the chaos computing elements enhances the robustness of chaos computing. [ABSTRACT FROM AUTHOR]

Based on an integration by parts formula for closed and convex subsets of a separable real Hilbert space with respect to a Gaussian measure, we first construct and identify the infinite dimensional analogue of the obliquely reflected Ornstein-Uhlenbeck process (perturbed by a bounded drift ) by means of a Skorokhod type decomposition. The variable oblique reflection at a reflection point of the boundary is uniquely described through a reflection angle and a direction in the tangent space (more precisely through an element of the orthogonal complement of the normal vector) at the reflection point. In case of normal reflection at the boundary of a regular convex set and under some monotonicity condition on , we prove the existence and uniqueness of a strong solution to the corresponding SDE. Subsequently, we consider an increasing sequence of closed and convex subsets of and the skew reflection problem at the boundaries of this sequence. We present concrete examples and obtain as a special case the infinite dimensional analogue of the -skew reflected Ornstein-Uhlenbeck process. [ABSTRACT FROM AUTHOR]

We review the level-crossing problem which includes the first-passage and escape problems as well as the theory of extreme values (the maximum, the minimum, the maximum absolute value and the range or span). We set the definitions and general results and apply them to one-dimensional diffusion processes with explicit results for the Brownian motion and the Ornstein-Uhlenbeck (OU) process. [ABSTRACT FROM AUTHOR]

We address the dynamics of a two-qubit system interacting with a classical dephasing environment driven by a Gaussian stochastic process. Upon introducing the concept of entanglement-preserving time, we compare the degrading effects of different environments, e.g. those described by Ornstein-Uhlenbeck (OU) or fractional noise. In particular, we consider pure Bell states and mixtures of Bell states and study the typical values of the entanglement-preserving time for both independent and common environments. We found that engineering environments towards fractional Gaussian noise is useful to preserve entanglement as well as to improve its robustness against noise. We also address entanglement sudden death by studying the sudden-death time as a function of the initial negativity. We found that: (i) the sudden-death time is bounded from below by an increasing function of the initial negativity, (ii) the sudden-death time depends only slightly on the process used to describe the environment and exhibits typicality. Overall, our results show that engineering the environment has only a slight influence over the entanglement sudden-death time, while it represents a valuable resource to increase the entanglement-preserving time, i.e. to maintain entanglement closer to the initial level for a longer interaction time. [ABSTRACT FROM AUTHOR]

The mean escape time of Brownian particles from a metastable potential well is investi-gated under the influence of a thermal time derivative Ornstein--Uhlenbeck noise, which can induce ballistic diffusion of force-free Brownian particles. Compared with the usual Ornstein--Uhlenbeck noise, some new characteristics are obtained. The Brownian parti-cles escape fast remarkably for a larger correlation time of the Ornstein-Uhlenbeck noise. The mean escape time increases as the correlation time decreases, which is contrary to the Ornstein-Uhlenbeck noise case. The escape time for derivative Ornstein-Uhlenbeck noise is longer than that of Ornstein-Uhlenbeck noise for small strength of noise and small correlation time of Ornstein-Uhlenbeck noise. These features are explained by the spectrum features of derivative Ornstein-Uhlenbeck noise and the corresponding dynamical effects. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a one-electron in a one-dimensional (1d) alloy with a correlated Ornstein-Uhlenbeck (OU) disorder distribution. The model considered here corresponds to an alloy with three types of atoms where the position of each atom is obtained using a stochastic rule based on the OU process. We analyze in detail the effect of this correlated disorder in the optical absorption spectrum and the level spacing statistics near the band center. Our results reveal a new collection of optical absorption peaks. We explain in details the appearance of each peak. Our calculations about the level spacing's distribution reveals a Poisson distribution thus contradicting previous statements about the existence of extended states in ternary electronic models with correlated disorder distribution. [ABSTRACT FROM AUTHOR]

This paper is devoted to the determination of the asymptotical optimal input for the estimation of the drift parameter in a partially observed but controlled fractional Ornstein-Uhlenbeck process. Large sample asymptotical properties of the Maximum Likelihood Estimator are deduced using Ibragimov-Khasminskii program and Laplace transform computations. [ABSTRACT FROM AUTHOR]

We study first passage problems of a class of reflected generalized Ornstein-Uhlenbeck processes without positive jumps. By establishing an extended Dynkin's formula associated with the process, we derive that the joint Laplace transform of the first passage time and an integral functional stopped at the time satisfies a truncated integro-differential equation. Two solvable examples are presented when the driven Lévy process is a drifted-Brownian motion and a spectrally negative stable process with index α ∈ (1, 2], respectively. Finally, we give two applications in finance. [ABSTRACT FROM AUTHOR]

The statistical properties of the Ornstein-Uhlenbeck (OU) process, a colored noise process, confirm the real noise statistics, since the real noise process has finite, nonzero correlation time. For this reason, it seems worthwhile to develop the estimation-theoretic scenarios of dynamical systems embedded in the colored noise environment as well. Importantly, the application of the Itô theory is not straightforward to the dynamical system in which the OU variable is a driving input. The augmented solution vector approach coupled with the Itô stochastic differential rule plays the pivotal role to develop the theory of the OU process-driven Duffing-van der Pol (DvdP) system of this paper. Notably, the noise analysis of the Duffing-van der Pol system, especially from the estimation-theoretic viewpoint, under the colored noise influence is not available yet in literature. Numerical experimentations with three different sets of data are demonstrated to examine the efficacy of analytical findings of this paper. The results of this paper will be of interest to noise scientists, especially research communities in systems and control, looking for the estimation-theoretic scenarios of the colored noise-driven "vector" stochastic differential system. [ABSTRACT FROM AUTHOR]

New stochastic models of thermodiffusion are constructed and their hydrodynamical limits are studied through a first-order Chapman–Enskog expansion. These models differ from earlier ones by taking into account all first-order contributions proportional to the temperature gradient and, thus, allow for both positive and negative Soret coefficients, in accordance with observations. [ABSTRACT FROM AUTHOR]

We consider an infinite-dimensional Ornstein–Uhlenbeck process on the spatial domain ]0,1[d driven by an additive nuclear or space–time white noise, and we study the approximation of this process at a fixed point in time. We determine the order of the minimal errors as well as asymptotically optimal algorithms, both of which depend on the spatial dimension d and on the decay of the eigenvalues of the driving Wiener process W in the case of nuclear noise. In particular, the optimal order is achieved by employing drift-implicit Euler schemes with non-uniform time discretizations, while uniform time discretizations turn out to be suboptimal in general. By means of non-asymptotic error bounds and by simulation experiments, we show that the asymptotic results are predictive for the actual errors already for time discretizations with a small number of points. [ABSTRACT FROM AUTHOR]

The synchronization of Stratonovich stochastic differential equations (SDE) with a one-sided dissipative Lipschitz drift and linear multiplicative noise is investigated by transforming the SDE to random ordinary differential equations (RODE) and synchronizing their dynamics. In terms of the original SDE, this gives synchronization only when the driving noises are the same. Otherwise, the synchronization is modulo exponential factors involving Ornstein–Uhlenbeck processes corresponding to the driving noises. Moreover, this occurs no matter how large the intensity coefficients of the noise. [ABSTRACT FROM AUTHOR]

This work investigates the global stability of cellular neural networks with distributive time delays and noise perturbations. Two types of noise perturbations are considered in the system, one is internal noise and another is external noise. We show that the deterministic system preserves the globally exponential stability when it is perturbed by the internal noise with small noise strength. On the other hand, the deterministic system can only preserve weaker stability when it is perturbed by the external noise. We also provide some numerical simulations to support our theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper deals with the regularity of an invariant measure μ associated to a class of generalized Ornstein–Uhlenbeck operators. Regularity here means that μ is absolutely continuous with respect to a properly chosen Gaussian reference measure σ on a separable Hilbert space H. Moreover, the square root of its Radon–Nikodym derivative ρ should belong to some directional Sobolev space $W^{1,2}_{(-A)^{-\gamma}} (H, \sigma)$. [ABSTRACT FROM AUTHOR]

We study the fluctuation limits of a class of superprocesses with dependent spatial motion on the real line, which give rise to some new Ornstein–Uhlenbeck processes with values of Schwartz distributions. [ABSTRACT FROM AUTHOR]

We prove the essential self-adjointness of Ornstein–Uhlenbeck operators perturbed by a dissipative drift and a potential with polynomial growth, by establishing certain gradient estimates and using Berezanskii's parabolic criterion. [ABSTRACT FROM AUTHOR]

A representation of a diffusion operator A on a certain space of right continuous trajectories in R[sup d] is derived. This operator A can be regarded as a flat copy of the OrnsteinUhlenbeck operator on the path space over a Riemannian manifold. The correspondence to E. P. Hsu's representation of the Ornstein-Uhlenbeck operator is shown. Furthermore, an alternative treatment of conditional expectations of A is given. [ABSTRACT FROM AUTHOR]

The motion of an inertial particle in a Gaussian random field is studied. This is a model for the phenomenon of preferential concentration, whereby inertial particles in a turbulent flow can correlate significantly. Mathematically the motion is described by Newton's second law for a particle on a 2-D torus, with force proportional to the difference between a background fluid velocity and the particle velocity itself. The fluid velocity is defined through a linear stochastic PDE of Ornstein-Uhlenbeck type. The properties of the model are studied in terms of the covariance of the noise which drives the stochastic PDE. Sufficient conditions are found for almost sure existence and uniqueness of particle paths, and for a random dynamical system with a global random attractor. The random attractor is illustrated by means of a numerical experiment, and the relevance of the random attractor for the understanding of particle distributions is highlighted. [ABSTRACT FROM AUTHOR]

The behavior of diffusively coupled Rössler oscillators parametrically perturbed with an Ornstein-Uhlenbeck noise is analyzed in terms of the degree of synchronization between the cells. A resonance-like behavior is found as a function of the noise correlation time, instead of the noise intensity as it occurs in the typical stochastic resonance. A power law scaling between the "optimum" correlation time with regard to synchronization and the deterministic time scale of the oscillators has been obtained, with an exponent that depends on the coupling strength. The behavior of diffusively coupled Rössler oscillators parametrically perturbed with an Ornstein-Uhlenbeck noise is analyzed in terms of the degree of synchronization between the cells. A resonance-like behavior is found as a function of the noise correlation time, instead of the noise intensity as it occurs in the typical stochastic resonance. A power law scaling between the "optimum" correlation time with regard to synchronization and the deterministic time scale of the oscillators has been obtained, with an exponent that depends on the coupling strength. [ABSTRACT FROM AUTHOR]

Existence and continuity results are obtained for self-intersection local time of S'(R[sup d])valued Ornstein-Uhlenbeck processes X[sub t] = T'[sub t]/X[sub 0] + ∫[sup t][sub 0] T'[sub t-s] dW[sub s], where X[sub 0] is Gaussian, W[sub t] is an S'(R[sup d])-Wiener process (independent of X[sub 0]), and T'[sub t] the adjoint of a semigroup T[sub t] on s(R[sup d]). Two types of covariance kernels for X[sub 0] and for W are considered: square tempered kernels and homogeneous random field kernels. The case where T[sub t] corresponds to the spherically symmetric α-stable process in R[sup d], α ∈ (0, 2], is treated in detail. The method consists in proving first results for self-intersection local times of the ingredient processes: W[sub t], T'[sub t]X[sub 0] and ∫[sub t][sub 0] T'[sub t-s] dWs, from which the results for X[sub t] are derived. As a by-product, a class of non-finite tempered measures on R[sup d] whose Fourier transforms are functions is identified. The tools are mostly analytical. [ABSTRACT FROM AUTHOR]

The melting of an ultrathin lubricant film is studied at friction between atomically flat surfaces. We take into account fluctuations of lubricant temperature, which are defined by the Ornstein-Uhlenbeck process. Phase diagrams and portraits are calculated for second- and first-order transitions (the melting of an amorphous and that of a crystalline lubricants, respectively). It is shown that, in the former case, a stick-slip friction domain, separating the regions of dry and sliding friction, appears. In the latter case, three domains of stick-slip friction arise, which are characterized by the transitions between dry, metastable and stable sliding friction. The increase in the correlation time of lubricant temperature fluctuations leads to increasing in the rubbing-surface temperature needed for realization of sliding friction. The stationary states, corresponding to dry, stable and metastable sliding friction, are reached as a result of damped oscillations. [ABSTRACT FROM AUTHOR]

We consider kinematics of a rigid body rotation about a fixed point. An angular velocity is considered to be a sum of deterministic and random vector-functions. We suppose that the components of the latter are statistically independent small stationary processes. The problem is, for example, important for the theory of a rotational Brownian motion or an error analysis of strapped-down inertial navigation systems. We construct a multiplicative perturbation theory so that the desired matrix is represented as a product of deterministic matrix by random perturbation one. The perturbation matrix is approximately found by the first terms of a matrizant. As example, we consider the constant deterministic velocity and the components of the colored random velocity to be white noises (the Brownian motion), the Ornstein-Uhlenbeck processes, or the time derivative of the Ornstein-Uhlenbeck processes ("green" noises). In the second case it is shown that a diffusion of an instanta-neous rotation angle appears only if the deterministic rotation is not zero. [ABSTRACT FROM AUTHOR]