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The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. In this article, we study the BBS when started from a random two-sided infinite particle configuration. For such a model, Ferrari et al. recently showed the invariance in distribution of Bernoulli product measures with density strictly less than

In this paper, time changes of the Brownian motions on generalized Sierpinski carpets including

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.

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.

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.

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

This study investigates the fractional (3 + 1)-dimensional equation for fluids with gas bubbles through conformable fractional operator, incorporating a noise term through the modified Sardar sub-equation method. This equation describes nonlinear wave propagation in compressible fluids containing distributed gas microbubbles, which is ubiquitous in biological tissues, petroleum engineering, acoustics, and fluid-structure interactions. The fractional derivative incorporates memory effects and anomalous dispersion, enhancing the model’s accuracy in reflecting real-world phenomena. The mathematical system is transformed into a nonlinear ordinary differential equation by applying a complex traveling wave transformation. We provide some innovative stochastic solutions through free physical parameters for the suggested model via Brownian motion process. The novel stochastic solutions investigated how noise affects frequencies in nonlinear systems such as biomedical ultrasonography, nuclear science, fluid-structure interaction, and more. To demonstrate the behavior of the offered stochastic solutions, a variety of profile graphs were created utilizing the Matlab release’s capabilities. The proposed methodology can ultimately be modified for use with a range of other models in applied science.

In conjunction with least squares method and generalized hat functions, we propose a new algorithm for stochastic Itô-Volterra integral equations. Firstly, the original problem is turned into solving a linear system of equations. Further, an efficient strategy is constructed to figure out the relevant coefficients of the linear system of equations. For computation purposes, throughout this paper, stochastic Itô integrals are transformed into conventional integrals using integration by parts formula. We also theoretically examine the convergence of the proposed approach. In the end, we provide two related examples to verify the reliability and accuracy of our proposed method. And in comparison with their numerical errors of the traditional block pulse method, the error of our presented approach is smaller.

A novel and unified solver technique is developed for handling a wide class of nonlinear systems of partial differential equations (NPDEs) that can be systematically reduced to the standard diffusing form with cubic nonlinearity. This canonical structure represents a broad spectrum of nonlinear evolution equations arising in nonlinear optics, superfluids, plasma physics, and quantum field theory. The proposed solver provides a robust analytical framework that efficiently transforms complex NPDEs into solvable ordinary differential forms by applying a proper wave transformation. Its adaptability allows for accurate extraction of solitary, periodic, and stochastic wave solutions under diverse boundary conditions. The solver is primarily used to study the stochastic δ -nonlinear Schrödinger equation (δ -NLSE), which incorporates random fluctuations from Brownian motion into nonlinear dispersive dynamics with δ -type localized perturbations. This application highlights the solver's ability to handle deterministic and stochastic nonlinearities, providing detailed insights into how noise and localized singularities affect the stability and propagation of nonlinear waves in complicated physical mediums. This work presents, for the first time, several analytical solutions to the δ -NLSE with Brownian noise. The results highlight the accuracy and efficiency of the proposed approach, emphasizing its applicability to address other intricate models in the natural sciences.