This paper discusses the weak pullback attractors for a damped stochastic fractional Schrödinger equation on [Formula: see text] with [Formula: see text]. By utilizing the stochastic Strichartz estimates and a stopping time technique argument, the existence and uniqueness of a global solution for the systems with the nonlinear term [Formula: see text] are proven. Furthermore, we define a mean random dynamical system due to the uniqueness of the solution, which has a unique weak pullback mean random attractor in [Formula: see text]. This result highlights the long-term dynamics of a broad class of stochastic fractional dispersion equations.
We propose a highly nonlinear mean-reverting stochastic model driven by stochastic correlation. Due to the absence of a tractable closed-form solution, we construct the truncated EulerMaruyama (EM) method to approximate the system dynamics and study the true and numerical solutions. Under suitable conditions, we establish that the numerical solutions converge to the true solution in probability as the step size approaches zero. Extensive simulation results are provided to illustrate the efficiency, stability, and practical applicability of the constructed method under various scenarios.
Stochastic control of delayed systems is a challenging research area. This paper investigates the near-optimal control problem for systems described by stochastic delayed differential equations with jumps, where both the delayed state and control influence the drift, diffusion and jump-diffusion terms, under a non-convex control domain. We establish necessary and sufficient conditions for near-optimality by employing the near-maximum condition on the H-function, which extends the Hamiltonian function in an integral sense. The main results rely on Ekeland's variational principle, stability properties of the state, and the first and second adjoint processes associated with the control variable. Finally, to demonstrate the applicability of our theoretical findings, we analyze a near-optimal advertising expenditure strategy aimed at minimizing the cost function in a delayed advertising model that incorporates social network effects.
In this paper, we investigate the random dynamics of a system consisting of N interacting stochastic partial differential equations (SPDEs) with a mean-field interaction, where the interacting potential is Lipschitz continuous and odd. Leveraging the mean-field structure, we decompose the system into its ensemble average and the fluctuation component. A Lyapunov-Perron method is then employed to establish the existence of a finite-dimensional random invariant manifold for large interaction. Further we give a mean-field limit approximation of the reduced system on the random invariant manifold. Our result shows that random dynamics of the N interacting particle system is determined by that of the ensemble average part of the system for large interaction and the approximating system is deterministic as N ->infinity. At last, our results are clearly illustrated by one example.
The fractional nonclassical diffusion equations with delay and nonlocal damping driven by additive noise is considered on the entire space & Ropf;(n). We mainly establish the upper semicontinuity of the pullback random attractors when noise intensity and time delay approach zero, respectively. It is worth emphasizing that we deal with the nonlocal term in Fourier space instead of introducing some new variables as in previous literatures.
In this paper, we study a class of stochastic differential equations modeling diffusive phenomena with state-dependent and distribution-dependent friction. In the small-mass regime, the dynamics are shown to be governed by the Smoluchowski-Kramers approximation. We obtain the limiting equation and characterize the resulting additional drift terms via solutions of associated Lyapunov and Sylvester matrix equations. Moreover, we establish the rate of convergence and extend the model to incorporate more general interaction mechanisms and noise structures.
Non-Gaussian colored noise and time delay are pivotal factors regulating the dynamical behaviors of neuronal systems, yet their combined influences on the state transition and regional stability of the FitzHugh-Nagumo (FHN) neural model remain insufficiently explored. To address the non-Markovian nature of the original system, we first convert it into an equivalent Markovian system by integrating the Unified Colored-Noise Approximation (UCNA) and small delay approximation theories. We then employ Mean First Exit Time (MFET) and First Escape Probability (FEP) to quantify the excited-to-resting state transition and introduce the Stochastic Basin of Attraction (SBA) to evaluate the stability of the excited region. Key results indicate that (i) The non-Gaussian colored noise intensity D, time delay tau, and deviation parameter q act as critical control parameters inducing the excited-to-resting transition; (ii) Increasing D, tau, or q could lead to a reduction in MFET and an increase in FEP, thereby enhancing the system's propensity for escaping from the excited region to the resting one; (iii) Stronger stochastic perturbations and longer time delays result in a shrinkage of the SBA size, which indicates a weakening of the stability of the excited region. This study provides new insights into the underlying regulatory mechanisms of noise and delay in neuronal dynamics, thus facilitating a deeper understanding of the complexity and dynamics of neuronal activity.
This paper studies a new class of damped fractional stochastic differential systems driven by L & eacute;vy process with impulsive effects. By employing the (alpha,gamma)-regularized family along with the fixed point technique, we studied the outcomes of approximate controllability for the proposed system by using the new type of control function. Finally, an example is provided to expound our theoretical conclusions.
Predicting critical transitions in complex systems, such as epileptic seizures in the brain, represents a major challenge in scientific research. The high-dimensional characteristics and hidden critical signals further complicate early-warning tasks. This study proposes a novel early-warning framework that integrates manifold learning with stochastic dynamical system modeling. Through systematic comparison, six methods including Diffusion Maps (DMs) are selected to construct low-dimensional representations. Based on these, a data-driven stochastic differential equation model is established to robustly estimate the probability evolution scoring function of the system. Building on this, a new Score Function (SF) indicator is defined by incorporating Schr & ouml;dinger bridge theory to quantify the likelihood of significant state transitions in the system. Experiments demonstrate that this indicator exhibits higher sensitivity and robustness in epilepsy prediction, enables earlier identification of critical points, and clearly captures dynamic features across various stages before and after seizure onset. This work provides a systematic theoretical framework and practical methodology for extracting early-warning signals from high-dimensional data.
Stochastic fluctuations play a crucial role in shaping the dynamics of infectious diseases such as those caused by coronaviruses. To understand these dynamics, it is essential to examine the effects of random perturbations on epidemic models. In this study, we formulate a stochastic Susceptible-Infectious-Recovered (SIR) model for coronavirus transmission, where the contact rate is subject to Levy noise. This approach captures discontinuous, high-intensity disturbances in transmission behavior. We first analyze the corresponding deterministic system. For the stochastic model driven by Levy noise, we establish the existence, uniqueness, and global positivity of solutions-fundamental prerequisites for meaningful biological analysis. We then derive a stochastic threshold parameter that governs the long-term fate of the infection. Using Lyapunov analysis and martingale techniques, we provide rigorous criteria for almost sure exponential extinction when this parameter falls below one, and for persistence in mean when it exceeds one. These results demonstrate how Levy-driven noise can fundamentally alter classical deterministic thresholds. Numerical simulations corroborate our theoretical findings: when the stochastic reproduction number is less than one, the disease dies out despite substantial stochastic perturbations; conversely, when it exceeds one, sustained fluctuations render control substantially more difficult. Collectively, this study offers fundamental theoretical insights into understanding and managing epidemic trajectories under the combined influence of continuous variability and discontinuous shocks.
In this paper, we study the asymptotic behavior for a class of slow–fast McKean–Vlasov stochastic differential equations subject to small noise perturbations. A distinctive feature of our model is that all coefficients depend on the probability distributions of both the slow component and the fast motion. By introducing the Poisson equation on Wasserstein space and applying the weak convergence approach, we establish a Freidlin–Wentzell-type uniform large deviation principle as the small noise parameter [Formula: see text] and the time scale parameter [Formula: see text].
A McKean–Vlasov stochastic differential equation subject to killing associated to a regularised non-conservative and path-dependent nonlinear parabolic partial differential equation is studied. The existence and pathwise uniqueness of a strong solution and the regularity properties of its sub-probability law are proved. The density of such a law may be seen as a weak solution of the considered PDE. The well-posedness of the associated particle system is also discussed.
The effect of multiplicative white noise on the resonance capture in non-isochronous systems with time-decaying pumping is investigated. It is assumed that the intensity of perturbations decays with time, and its frequency is asymptotically constant. The occurrence of attractive solutions with an amplitude close to the resonant value and a phase synchronized with the excitation is considered. The persistence of such a regime in a stochastically perturbed system is analyzed. By combining the averaging method and the construction of suitable stochastic Lyapunov functions, conditions are derived that guarantee the stochastic stability of the resonant modes on infinite or asymptotically large time intervals. The proposed theory is applied to the Duffing oscillator with decaying parametric excitation and noise.
In this paper, we introduce and study the convergence of new Carath & eacute;odory's approximate solution for one-dimensional alpha,beta-doubly perturbed stochastic differential equations (DPSDEs) with parameters alpha < 1 and beta < 1 such that |rho| < 1, where rho :=( alpha beta)/ ((1-alpha)(1-beta)). Under Lipschitz's condition on the coefficients, we establish the Lp-convergence of the Carath & eacute;odory approximate solution uniformly in time, for all p >= 2. As a consequence, and relying only on our scheme, we obtain the existence and uniqueness of strong solution for alpha,beta-DPSDEs. Furthermore, an extension to a class of non-Lipschitz coefficients also studied. Our results improve earlier work by Mao et al. [Approximate solutions for a class of doubly perturbed stochastic differential equations, Adv. Differential Equations 2018(1) (2018) 1-17].
Let M be a compact manifold equipped with a pair of complementary foliations, say horizontal H and vertical & Vscr;. In Melo, Morgado and Ruffino [Decomposition of stochastic flows generated by Stratonovich SDEs with jumps, Discrete Contin. Dyn. Syst. Ser. B 21(9) (2016) 3209-3218] it is proved that if a semimartingale X-t has a finite number of jumps in compact intervals then, up to a stopping time tau, a stochastic flow of local diffeomorphisms in M driven by X-t can be decomposed into a process in the Lie group of diffeomorphisms which fix the leaves of H composed with a process in the Lie group of diffeomorphisms which fix the leaves of & Vscr;. Dynamics at the discontinuities of X-t here are interpreted in the Marcus sense as in Kurtz, Pardoux and Protter [Stratonovich stochastic differential equations driven by general semimartingales, Ann. Inst. Henri Poincar Probab. Stat. 31(2) (1995) 351-377]. Here, we enlarge the scope of this geometric decomposition and consider flows driven by arbitrary semimartingales with jumps and show explicit equations for each component. Our technique is based in an extension of the It & ocirc;-Ventzel-Kunita formula for stochastic flows with jumps. Geometrical and other topological obstructions for the decomposition are also considered: e.g., an index of attainability is introduced to measure the complexity of the dynamics with respect to the pair of foliations.
In this paper, we consider the 2D periodic stochastic Nernst-Planck-Navier-Stokes equations with body forces perturbed by multiplicative white noise. We first transform the stochastic Nernst-Planck-Navier-Stokes system into the deterministic system and address the problem of global well-posedness of the solution. Then, we generate a corresponding random dynamical system and dedicate to proving the existence of a compact random attractor for such random dynamical system. Furthermore, upper semicontinuity of the random attractor is established when the noise intensity approaches zero.
We propose a theory of unimodal maps perturbed by a heteroscedastic Markov chain noise and experiencing another heteroscedastic noise due to uncertain observation. We address and treat the filtering problem showing that by collecting more and more observations, one would predict the same distribution for the state of the underlying Markov chain no matter one's initial guess. Moreover, we give other limit theorems, emphasizing in particular concentration inequalities and extreme value and Poisson distributions. Our results apply to a family of maps arising from a model of systemic risk in finance.
In this paper, we consider semi-Markov processes whose transition times and transition probabilities depend on a small parameter epsilon. We are interested in the behavior of a process X-t(epsilon) at times t = t(epsilon) that depend on the value of the parameter. This behavior depends on how the point (1/epsilon,t(epsilon)) approaches infinity. We introduce the notion of complete asymptotic regularity (a certain asymptotic condition on transition probabilities and transition times), originally developed for parameter-dependent Markov chains, which ensures the existence of the metastable distribution for each initial point and a given time scale t(epsilon). The result may be viewed as a generalization of the ergodic theorem to the case of parameter-dependent semi-Markov processes.
This work is devoted to studying asymptotic behaviors for Volterra type McKean-Vlasov stochastic differential equations with small noise. By applying the weak convergence approach, we establish the large and moderate deviation principles. In addition, we obtain the central limit theorem and find the Volterra integral equation satisfied by the limiting process, which involves the Lions derivative of the drift coefficient.
