Explore the vast range of titles available.

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.

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.

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.

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.

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.

We show that the linear drift of the Brownian motion on the universal cover of a closed connected smooth Riemannian manifold is

Uncertainty is a significant challenge in tsunami hazard analysis. Tsunami heights are affected by complex factors and change constantly during propagation. The heights of tsunami have random characteristics. This study proposes that the water depths (related to seabed topography) are the most important factors that affect tsunami height. But across the globe, a considerable area of seabed topography has not been measured. So it is necessary to use the method of uncertainty to consider the water depth. The Wiener process is utilized to quantify the random changes of the water depth, which can better describe the situation that water depths change in a non-monotonic way. Considering the uncertainty of water depth, a Weiner process-based probabilistic model was established for predicting the maximum tsunami height, which is different from the maximum tsunami height deterministic or stochastic model previously studied with higher prediction efficiency and good prediction accuracy. The probability distribution of maximum tsunami heights was calculated using the stochastic model. The mean value of the maximum tsunami heights was very similar to the average value of 165 actual observations of maximum tsunami heights collected from 1997 to 2017.

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.

It is known that certain one-dimensional nearest-neighbor random walks in i.i.d. random space-time environments have diffusive scaling limits. Here, in the continuum limit, the random environment is represented by a ‘stochastic flow of kernels’, which is a collection of random kernels that can be loosely interpreted as the transition probabilities of a Markov process in a random environment. The theory of stochastic flows of kernels was first developed by Le Jan and Raimond, who showed that each such flow is characterized by its Our main result gives a graphical construction of general Howitt-Warren flows, where the underlying random environment takes on the form of a suitably marked Brownian web. This extends earlier work of Howitt and Warren who showed that a special case, the so-called ‘erosion flow’, can be constructed from two coupled ‘sticky Brownian webs’. Our construction for general Howitt-Warren flows is based on a Poisson marking procedure developed by Newman, Ravishankar and Schertzer for the Brownian web. Alternatively, we show that a special subclass of the Howitt-Warren flows can be constructed as random flows of mass in a Brownian net, introduced by Sun and Swart. Using these constructions, we prove some new results for the Howitt-Warren flows. In particular, we show that the kernels spread with a finite speed and have a locally finite support at deterministic times if and only if the flow is embeddable in a Brownian net. We show that the kernels are always purely atomic at deterministic times, but, with the exception of the erosion flows, exhibit random times when the kernels are purely non-atomic. We moreover prove ergodic statements for a class of measure-valued processes induced by the Howitt-Warren flows. Our work also yields some new results in the theory of the Brownian web and net. In particular, we prove several new results about coupled sticky Brownian webs and about a natural coupling of a Brownian web with a Brownian net. We also introduce a ‘finite graph representation’ which gives a precise description of how paths in the Brownian net move between deterministic times.