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We study the long-time behavior of nonlinear stochastic evolution equations in a separable Hilbert space driven by a Q-Wiener process. The linear part of the equation is generated by a strongly continuous semigroup with an exponential dichotomy, which provides fixed rates of decay and growth. The nonlinear drift and diffusion terms are globally Lipschitz and become small as time tends to infinity. Our main result shows that under these conditions, the mean-square Lyapunov exponents of the nonlinear system coincide with those of the linear part. In other words, nonlinear stochastic perturbations that decay in time do not change the main growth or decay rates of solutions in the mean-square sense. This result provides simple and verifiable criteria ensuring that the long-time Lyapunov behavior of the nonlinear stochastic equation is fully determined by the linear semigroup, even in the presence of time-dependent stochastic perturbations.

We construct weak solutions to the McKean–Vlasov SDE d X ( t ) = b ( X ( t ) , d L X ( t ) d x ( X ( t ) ) ) d t + σ ( X ( t ) , d L X ( t ) d t ( X ( t ) ) ) d W ( t ) on ℝ d for possibly degenerate diffusion matrices σ with X(0) having a given law, which has a density with respect to Lebesgue measure, dₓ. Here, 𝓛 X(t) denotes the law of X(t). Our approach is to first solve the corresponding non-linear Fokker–Planck equations and then use the well-known superposition principle to obtain weak solutions of the above SDE.

Motivated by our recent proposal linking stochastic processes and Schrödinger equation, we use the Euler–Maruyama technique to show that a class of Wiener processes exist that are obtained by computing an arbitrary positive power of them. This can be accomplished with a proper set of definitions that makes meaningful the realization at discrete times of these processes and make them computable. Standard results from Itō calculus for integer powers hold as we are just extending them. We provide the results from a Monte Carlo simulation with a large number of samples. We yield evidence for the existence of these processes by recovering from them the standard Brownian motion we started with after power elevation. The perfect coincidence of the numerical results we obtained is a clear evidence of existence of these processes. This could pave the way to a generalization of the concepts of stochastic integral and relative process.

We consider a semilinear parabolic PDE driven by additive noise. The equation is discretized in space by a standard piecewise linear finite element method. We show that the orthogonal expansion of the finite-dimensional Wiener process, that appears in the discretized problem, can be truncated severely without losing the asymptotic order of the method, provided that the kernel of the covariance operator of the Wiener process is smooth enough. For example, if the covariance operator is given by the Gauss kernel, then the number of terms to be kept is the quasi-logarithm of the number of terms in the original expansion. Then one can reduce the size of the corresponding linear algebra problem enormously and hence reduce the computational complexity, which is a key issue when stochastic problems are simulated.

This article systematically investigates the inverse Gaussian (IG) process as an effective degradation model. The IG process is shown to be a limiting compound Poisson process, which gives it a meaningful physical interpretation for modeling degradation of products deteriorating in random environments. Treated as the first passage process of a Wiener process, the IG process is flexible in incorporating random effects and explanatory variables that account for heterogeneities commonly observed in degradation problems. This flexibility makes the class of IG process models much more attractive compared with the Gamma process, which has been thoroughly investigated in the literature of degradation modeling. The article also discusses statistical inference for three random effects models and model selection. It concludes with a real world example to demonstrate the applicability of the IG process in degradation analysis. Supplementary materials for this article are available online.

The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy In this paper, we employ the Brownian Gibbs property to make a close comparison between the Airy line ensemble’s curves after affine shift and Brownian bridge, proving the finiteness of a superpolynomially growing moment bound on Radon-Nikodym derivatives. We also determine the value of a natural exponent describing in Brownian last passage percolation the decay in probability for the existence of several near geodesics that are disjoint except for their common endpoints, where the notion of ‘near’ refers to a small deficit in scaled geodesic energy, with the parameter specifying this nearness tending to zero. To prove both results, we introduce a technique that may be useful elsewhere for finding upper bounds on probabilities of events concerning random systems of curves enjoying the Brownian Gibbs property. Several results in this article play a fundamental role in a further study of Brownian last passage percolation in three companion papers (Hammond 2017a,b,c), in which geodesic coalescence and geodesic energy profiles are investigated in scaled coordinates.

It is necessary to utilize certain stochastic methods while finding the soliton solutions since several physical systems are by their very nature stochastic. By adding randomness into the modeling process, researchers gain deeper insights into the impact of uncertainties on soliton evolution, stability, and interaction. In the realm of dynamics, deterministic models often encounter limitations, prompting the incorporation of stochastic techniques to provide a more comprehensive framework. Our attention was directed towards the short-wave intermediate dispersive variable (SIdV) equation with the Wiener process. By integrating advanced methodologies such as the modified Kudrayshov technique (KT), the generalized KT, and the sine-cosine method, we delved into the exploration of diverse solitary wave solutions. Through those sophisticated techniques, a spectrum of the traveling wave solutions was unveiled, encompassing both the bounded and singular manifestations. This approach not only expanded our understanding of wave dynamics but also shed light on the intricate interplay between deterministic and stochastic processes in physical systems. Solitons maintained stable periodicity but became vulnerable to increased noise, disrupting predictability. Dark solitons obtained in the results showed sensitivity to noise, amplifying variations in behavior. Furthermore, the localized wave patterns displayed sharp peaks and periodicity, with noise introducing heightened fluctuations, emphasizing stochastic influence on wave solutions.

Tool condition monitoring can be employed to ensure safe and full utilization of the cutting tool. Hence, remaining useful life (RUL) prediction of a cutting tool is an important issue for an effective high-speed milling process-monitoring system. However, it is difficult to establish a mechanism model for the life decreasing process owing to the different wear rates in various stages of cutting tool. This study proposes a three-stage Wiener-process-based degradation model for the cutting tool wear estimation and remaining useful life prediction. Tool wear stages classification and RUL prediction are jointly addressed in this work in order to take full advantage of Wiener process, as this three-stage Wiener process definitely constitutes to describe the degradation processes at different wear stages, based on which the overall useful life can be accurately obtained. The numerical results obtained using extensive experiment indicate that the proposed model can effectively predict the cutting tool’s remaining useful life. Empirical comparisons show that the proposed model performs better than existing models in predicting the cutting tool RUL.

In this paper the authors study mesoscopic fluctuations for Dyson's Brownian motion with \\beta =2. Dyson showed that the Gaussian Unitary Ensemble (GUE) is the invariant measure for this stochastic evolution and conjectured that, when starting from a generic configuration of initial points, the time that is needed for the GUE statistics to become dominant depends on the scale we look at: The microscopic correlations arrive at the equilibrium regime sooner than the macrosopic correlations. The authors investigate the transition on the intermediate, i.e. mesoscopic, scales. The time scales that they consider are such that the system is already in microscopic equilibrium (sine-universality for the local correlations), but have not yet reached equilibrium at the macrosopic scale. The authors describe the transition to equilibrium on all mesoscopic scales by means of Central Limit Theorems for linear statistics with sufficiently smooth test functions. They consider two situations: deterministic initial points and randomly chosen initial points. In the random situation, they obtain a transition from the classical Central Limit Theorem for independent random variables to the one for the GUE.

With the popularity of various social media, the propagation of rumors is becoming a social threat. Here, the proposed mathematical model signifies the dynamics of rumor propagation on social media with the influence of counter-rumor spreaders in regulating the transmission process as well as controlling its harmful effect. The total number of users is divided into four categories: (i) newcomer, (ii) spreaders, (iii) counter-rumor spreaders, iv ) stiflers. The spreading threshold ( R 0 ) of rumor transmission regulates the condition of the prevalence of rumor. ( R 0 ) < 1 assures the stability of rumor-free state, while ( R 0 ) > 1 assures that one prevailing state exists uniquely with stable nature. Condition for global stability of prevailing state for deterministic system is derived. Subsequently, the corresponding stochastic model demonstrates the effect of random external factors (Wiener process) on rumor propagation dynamics. The global existence and uniqueness of the solution are established to study the asymptotic behavior of that solution around the steady-states. We have also compared the persistence criterion of rumor propagation for the modified system with the deterministic system and derived the condition for the extinction of rumor. Furthermore, scatter plots indicate the significant impact of parameters and numerical simulations are presented to validate the analytical studies. Numerical results assure that environmental noise plays a significant role in suppressing rumor propagation.