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Authors:Xiangqian Yan, Wei Yan, Yajuan Zhao, Meihua Yang Abstract: Stochastics and Dynamics, Ahead of Print. This paper is devoted to studying the convergence problem of free reduced Ostrovsky equation in Fourier–Lebesgue spaces with rough data and the stochastic continuity of free reduced Ostrovsky equation in Fourier–Lebesgue spaces with random data. On the one hand, we establish the pointwise convergence related to the free reduced Ostrovsky equation in Fourier–Lebesgue spaces [math] with rough data. In particular, we show that [math] is the necessary condition for the maximal function estimate in [math], which means that [math] is optimal for rough data. On the other hand, we present the stochastic continuity of free reduced Ostrovsky equation at [math] in Fourier–Lebesgue spaces [math] with random data. Citation: Stochastics and Dynamics PubDate: 2022-07-30T07:00:00Z DOI: 10.1142/S0219493723500016

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Authors:Tomás Caraballo, José A. Langa, Alexandre N. Carvalho, Alexandre N. Oliveira-Sousa Abstract: Stochastics and Dynamics, Ahead of Print. In this work, we study the continuity and topological structural stability of attractors for nonautonomous random differential equations obtained by small bounded random perturbations of autonomous semilinear problems. First, we study the existence and permanence of unstable sets of hyperbolic solutions. Then, we use this to establish the lower semicontinuity of nonautonomous random attractors and to show that the gradient structure persists under nonautonomous random perturbations. Finally, we apply the abstract results in a stochastic differential equation and in a damped wave equation with a perturbation on the damping. Citation: Stochastics and Dynamics PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S021949372240024X

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Authors:Junfeng Liu, Zhi Wang, Zengwu Wang Abstract: Stochastics and Dynamics, Ahead of Print. In this paper, we study a class of space-time fractional Anderson model driven by multiplicative Gaussian noise which is white/colored in time and has the covariance of a fractional Brownian motion with Hurst index [math] in space. We prove the existence of the solution in the Skorohod sense and obtain the upper and lower bounds for the [math]th moments for all [math]. Then we can prove that solution of this equation in the Skorohod sense is weakly intermittent. We also deduce the Hölder continuity of the solution with respect to the time and space variables. Citation: Stochastics and Dynamics PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S021949372350003X

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Authors:Mojtaba Kiyanpour, Bijan Z. Zangeneh, Ruhollah Jahanipur Abstract: Stochastics and Dynamics, Ahead of Print. In this paper, we establish the existence and uniqueness of the mild solution for stochastic Volterra equation with a non-self-adjoint operator. The specific Volterra equation that we consider is a generalization of the fractional differential equation. To obtain the mild solution for the case of multiplicative problem, the resolvent property of the linear perturbation of a sectorial operator will be considered. Citation: Stochastics and Dynamics PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S0219493723500041

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Authors:Oleksiy Kapustyan, Oleksandr Misiats, Oleksandr Stanzhytskyi Abstract: Stochastics and Dynamics, Ahead of Print. In this paper, we study the conditions for the existence of strong solutions (both local and global) for stochastic bidomain equations. To this end, we use a priori energy estimates and Serrin-type theorems. We further address the asymptotic behavior of the solutions, which includes the analysis of small stochastic perturbations and large deviations. In a separate section we specify the support of the invariant measure, whose existence was established in [M. Hieber, O. Misiats and O. Stanzhytskyi, On the bidomain equations driven by stochastic forces, Discrete Contin. Dyn. Syst. 40(11) (2020) 6159–6177]. Citation: Stochastics and Dynamics PubDate: 2022-07-25T07:00:00Z DOI: 10.1142/S0219493722500277

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Authors:Hui Jiang, Yajuan Pan Abstract: Stochastics and Dynamics, Ahead of Print. In this paper, we study explicitly the deviation properties, including the deviation inequalities and Cramér-type moderate deviations, for some quadratic functionals of linear self-attracting diffusion process. As applications, Cramér-type moderate deviations for the log-likelihood ratio process and drift parameter estimator are obtained. The main methods consist of the deviation inequalities and Cramér-type moderate deviations for multiple Wiener–Itô integrals, as well as the asymptotic analysis techniques. Citation: Stochastics and Dynamics PubDate: 2022-07-25T07:00:00Z DOI: 10.1142/S0219493722500289

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Authors:Fabian A. Harang, Samy Tindel, Xiaohua Wang Abstract: Stochastics and Dynamics, Ahead of Print. We extend the recently developed rough path theory for Volterra equations from [F. Harang and S. Tindel, Volterra equations driven by rough signals, Stoch. Process. Appl. 142 (2021) 34–78] to the case of more rough noise and/or more singular Volterra kernels. It was already observed in [F. Harang and S. Tindel, Volterra equations driven by rough signals, Stoch. Process. Appl. 142 (2021) 34–78] that the Volterra rough path introduced there did not satisfy any geometric relation, similar to that observed in classical rough path theory. Thus, an extension of the theory to more irregular driving signals requires a deeper understanding of the specific algebraic structure arising in the Volterra rough path. Inspired by the elements of “non-geometric rough paths” developed in [M. Gubinelli, Ramification of rough paths, J. Differential Equations 248 (2010) 693–721; M. Hairer and D. Kelly, Geometric versus non-geometric rough path, Ann. Inst. Henri Poincaré-Probab. Stat. 51 (2015) 207–251], we provide a simple description of the Volterra rough path and the controlled Volterra process in terms of rooted trees, and with this description we are able to solve rough Volterra equations driven by more irregular signals. Citation: Stochastics and Dynamics PubDate: 2022-07-20T07:00:00Z DOI: 10.1142/S0219493723500028

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Authors:David Nualart, Ercan Sönmez Abstract: Stochastics and Dynamics, Ahead of Print. In this paper, we show the existence and uniqueness of a solution for a stochastic differential equation driven by an additive noise which is the sum of two fractional Brownian motions with different Hurst parameters. The proofs are based on the techniques of fractional calculus and Girsanov theorem. In particular, we show that the regularization effect of the fractional Brownian motion with the smaller Hurst index dominates. A key challenge in this paper is to extend and apply the Girsanov theorem for two noises given by the sum of two (dependent) fractional Brownian motions by using profound techniques of fractional operator theory. Citation: Stochastics and Dynamics PubDate: 2022-07-12T07:00:00Z DOI: 10.1142/S0219493722500290

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Authors:Shuo Yan Abstract: Stochastics and Dynamics, Ahead of Print. We consider a two-dimensional Hamiltonian system perturbed by a small diffusion term, whose coefficient is state-dependent and nondegenerate. As a result, the process consists of the fast motion along the level curves and slow motion across them. On finite time intervals, the large deviation principle applies, while on time scales that are inversely proportional to the size of the perturbation, the averaging principle holds, i.e. the projection of the process onto the Reeb graph converges to a Markov process. In our paper, we consider the intermediate time scales and prove the large deviation principle, with the action functional determined in terms of the averaged process on the graph. Citation: Stochastics and Dynamics PubDate: 2022-06-29T07:00:00Z DOI: 10.1142/S0219493722500253

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Authors:Franco Flandoli, Benjamin Gess, Francesco Grotto Abstract: Stochastics and Dynamics, Ahead of Print. A new mechanism leading to a random version of Burgers’ equation is introduced: it is shown that the Totally Asymmetric Exclusion Process in discrete time (TASEP) can be understood as an intrinsically stochastic, non-entropic weak solution of Burgers’ equation on [math]. In this interpretation, the appearance of randomness in the Burgers’ dynamics is caused by random additions of jumps to the solution, corresponding to the random effects in TASEP. Citation: Stochastics and Dynamics PubDate: 2022-06-27T07:00:00Z DOI: 10.1142/S0219493722400238

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Authors:Menghui Li, Jinchun He, Haoyuan Xu, Meihua Yang Abstract: Stochastics and Dynamics, Ahead of Print. In this paper, we consider a fractional Laplacian system with saturable nonlinearity. Under some assumptions on the parameters and potential functions, we obtain the existence and concentration behavior of the positive ground state solution by variational methods. Citation: Stochastics and Dynamics PubDate: 2022-06-20T07:00:00Z DOI: 10.1142/S0219493722500265

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Authors:Thiago Bomfim, Rongrong Huo, Paulo Varandas, Yun Zhao Abstract: Stochastics and Dynamics, Ahead of Print. The space of asymptotically additive potentials is a Banach space and, identifying any two pairs of sequences in terms of their limiting behavior, the quotient space [math] is endowed with a vector space structure. A recent correspondence between additive potentials and classes of asymptotically additive potentials allows us to prove that: (i) the class of asymptotically additive potentials having a unique maximizing measure forms a Baire residual subset of [math], (ii) for transitive hyperbolic homeomorphisms with local product structure, the elements in [math] for which every maximizing measure has full support form a Baire residual subset of [math], and that (iii) for expanding and Anosov maps, the elements in [math] whose unique maximizing measure has zero metric entropy form a Baire residual set of [math]. Further results stating that the maximizing measure is periodic and a number of applications are also discussed. Citation: Stochastics and Dynamics PubDate: 2022-06-10T07:00:00Z DOI: 10.1142/S0219493722500241

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Authors:K. Abdelhadi, N. Khelfallah Abstract: Stochastics and Dynamics, Ahead of Print. We study a backward SDE driven by a jump Markov process (BSDEJ for short) whose generator may be locally Lipschitz or of logarithmic growth in [math]-variables. The existence, uniqueness and stability theorems to such BSDEJs are established. We essentially approximate the initial problem by constructing a suitable sequence of BSDEJs with globally Lipschitz generators for which the existence and uniqueness of solutions hold. By passing to the limits, we show the existence and uniqueness of solutions to the original problems. We apply our main results to prove the existence of a unique solution to the Kolmogorov equation of the Markov process. Citation: Stochastics and Dynamics PubDate: 2022-05-18T07:00:00Z DOI: 10.1142/S0219493722500216

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Authors:Gregory Amali Paul Rose, Murugan Suvinthra, Krishnan Balachandran Abstract: Stochastics and Dynamics, Ahead of Print. This paper aims to establish the central limit theorem and moderate deviation principle for the stochastic Kuramoto–Sivashinsky equation driven by multiplicative noise on a bounded domain. The moderate deviation principle is investigated using the weak convergence approach based on a variational representation for expected values of positive functionals of the Brownian motion. The approach relies on proving basic qualitative properties of controlled versions of the original stochastic partial differential equation which is under consideration. Citation: Stochastics and Dynamics PubDate: 2022-05-18T07:00:00Z DOI: 10.1142/S021949372250023X

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Authors:Wilfried Grecksch, Hannelore Lisei Abstract: Stochastics and Dynamics, Ahead of Print. In this paper, we study the existence of the solution of a linear SPDE driven by a multiplicative multifractional Brownian motion. Moreover, we study an optimal control problem with a linear quadratic objective functional involving the solution of the studied SPDE. We prove the existence and uniqueness of the optimal control. Citation: Stochastics and Dynamics PubDate: 2022-05-10T07:00:00Z DOI: 10.1142/S0219493722400202

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Authors:Paulo Henrique Da Costa, Michael A. Högele, Paulo R. Ruffino Abstract: Stochastics and Dynamics, Ahead of Print. This paper refines the classical notion of a stochastic D-bifurcation to the respective family of n-point motions for homogeneous Markovian stochastic semiflows, such as stochastic Brownian flows of homeomorphisms, and their generalizations. This notion essentially detects at which level the support of the invariant measure of the k-point bifurcation has more than one connected component. Stochastic Brownian flows and their invariant measures were shown by Kunita (1990) to be rigid, in the sense of being uniquely determined by the [math]-and [math]-point motions. Hence, only stochastic n-point bifurcation of level [math] or [math] can occur. For general homogeneous stochastic Markov semiflows this turns out to be false. This paper constructs minimal examples of where this rigidity is false in general on finite space and studies the complexity of the resulting n-point bifurcations. Citation: Stochastics and Dynamics PubDate: 2022-05-10T07:00:00Z DOI: 10.1142/S0219493722400214

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Authors:Yarong Liu, Yejuan Wang, Tomás Caraballo Abstract: Stochastics and Dynamics, Ahead of Print. We consider stochastic 2D-Stokes equations with unbounded delay in fractional power spaces and moments of order [math] driven by a tempered fractional Brownian motion (TFBM) [math] with [math] and [math]. First, the global existence and uniqueness of mild solutions are established by using a new technical lemma for stochastic integrals with respect to TFBM in the sense of [math]th moment. Moreover, based on the relations between the stochastic integrals with respect to TFBM and fractional Brownian motion, we show the continuity of mild solutions in the case of [math], [math] or [math], [math]. In particular, we obtain [math]th moment Hölder regularity in time and [math]th polynomial stability of mild solutions. This paper can be regarded as a first step to study the challenging model: stochastic 2D-Navier–Stokes equations with unbounded delay driven by tempered fractional Gaussian noise. Citation: Stochastics and Dynamics PubDate: 2022-05-10T07:00:00Z DOI: 10.1142/S0219493722500228

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Authors:Obayda Assaad, Julie Gamain, Ciprian A. Tudor Abstract: Stochastics and Dynamics, Ahead of Print. We study the quadratic variations (in time and in space) of the solution to the stochastic wave equation driven by the space-time white noise. We give their limit (almost surely and in [math]) and we prove that these variations satisfy, after a proper renormalization, a Central Limit Theorem. We apply the quadratic variation to define and analyze estimators for the drift parameter of the wave equation. Citation: Stochastics and Dynamics PubDate: 2022-04-29T07:00:00Z DOI: 10.1142/S0219493722400147

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Authors:Tomás Caraballo, Francisco Morillas, José Valero Abstract: Stochastics and Dynamics, Ahead of Print. In this paper, we study a stochastic system of differential equations with nonlocal discrete diffusion. For two types of noises, we study the existence of either positive or probability solutions. Also, we analyze the asymptotic behavior of solutions in the long term, showing that under suitable assumptions they tend to a neighborhood of the unique deterministic fixed point. Finally, we perform numerical simulations and discuss the application of the results to life tables for mortality in Spain. Citation: Stochastics and Dynamics PubDate: 2022-04-18T07:00:00Z DOI: 10.1142/S0219493722400172

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Authors:Tuan Anh Phan, Shuxun Wang, Jianjun Paul Tian Abstract: Stochastics and Dynamics, Ahead of Print. In this paper, we analyze a new Ito stochastic differential equation model for untreated human glioblastomas. The model was the best fit of the average growth and variance of 94 pairs of a data set. We show the existence and uniqueness of solutions in the positive spatial domain. When the model is restricted in the finite domain [math], we show that the boundary point 0 is unattainable while the point [math] is reflecting attainable. We prove there is a unique ergodic stationary distribution for any non-zero noise intensity, and obtain the explicit probability density function for the stationary distribution. By using Brownian bridge, we give a representation of the probability density function of the first passage time when the diffusion process defined by a solution passes the point [math] firstly. We carry out numerical studies to illustrate our analysis. Our mathematical and numerical analysis confirms the soundness of our randomization of the deterministic model in that the stochastic model will set down to the deterministic model when the noise intensity approaches zero. We also give physical interpretation of our stochastic model and analysis. Citation: Stochastics and Dynamics PubDate: 2022-03-21T07:00:00Z DOI: 10.1142/S0219493722500198

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Authors:Oussama Elbarrimi Abstract: Stochastics and Dynamics, Ahead of Print. In this paper, we consider multidimensional mean-field stochastic differential equations where the coefficients depend on the law in the form of a Lebesgue integral with respect to the measure of the solution. Under the pathwise uniqueness property, we establish various strong stability results. As a consequence, we give an application for optimal control of diffusions. Namely, we propose a result on the approximation of the solution associated to a relaxed control. Citation: Stochastics and Dynamics PubDate: 2022-03-15T07:00:00Z DOI: 10.1142/S0219493722500204

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Authors:Wenya Wang, Zhongkai Guo Abstract: Stochastics and Dynamics, Ahead of Print. In this paper, a class of Itô–Doob stochastic fractional differential equations (Itô–Doob SFDEs) models are discussed. Using the time scale transformation method, we consider the averaging principle of the transformed equations and establish the relevant results. At the same time, we find that the optimal index for the original Itô–Doob SFDEs can be determined, the selection of such index is similar to the classical stochastic differential equations model. Citation: Stochastics and Dynamics PubDate: 2022-02-25T08:00:00Z DOI: 10.1142/S0219493722500186

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Authors:A. Lahrouz, A. Settati, M. Jarroudi, H. Mahjour, M. Fatini, M. Merzguioui, A. Tridane Abstract: Stochastics and Dynamics, Ahead of Print. In this paper, we revisit the classical SIR epidemic model by replacing the simple bilinear transmission rate by a nonlinear one. Our results show that in the presence of environmental fluctuations represented by Brownian motion and that mainly act on the transmission rate, the generalized non-concave force of infection adopted here, greatly affects the long-time behavior of the epidemic. Employing the Markov semigroup theory, we prove that the model solutions do not admit a unique stationary distribution but converge to 0 in [math]th moment for any [math]. Furthermore, we prove that the disease extinguishes asymptotically exponentially with probability 1 without any restriction on the model parameters and we also determine the rate of convergence. This is an unexpected qualitative behavior in comparison with the existing literature where the studied epidemic models have a threshold dynamics behavior. It is also a very surprising behavior regarding the deterministic counterpart that can exhibit a rich qualitative dynamical behaviors such as backward bifurcation and Hopf bifurcation. On the other hand, we show by several numerical simulations that as the intensity of environmental noises becomes sufficiently small, the epidemic tends to persist for a very long time before dying out from the host population. To solve this problem and to be able to manage the pre-extinction period, we construct a new process in terms of the number of infected and recovered individuals which admits a unique invariant stationary distribution. Finally, we discuss the obtained analytical results through a series of numerical simulations. Citation: Stochastics and Dynamics PubDate: 2022-01-26T08:00:00Z DOI: 10.1142/S0219493722500162

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Authors:Moustapha Dieye, Amadou Diop, Mark A. Mckibben Abstract: Stochastics and Dynamics, Ahead of Print. In this paper, we study the existence and continuous dependence on coefficients of mild solutions for first-order McKean–Vlasov integrodifferential equations with delay driven by a cylindrical Wiener process using resolvent operator theory and Wasserstein distance. Under the situation that the nonlinear term depends on the probability distribution of the state, the existence and uniqueness of solutions are established. An example illustrating the general results is included. Citation: Stochastics and Dynamics PubDate: 2022-01-26T08:00:00Z DOI: 10.1142/S0219493722500174