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We analyze a class of nonlinear partial differential equations (PDEs) defined on

We prove that for Bernoulli percolation on

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

The goal of the Entropy and the Quantum schools has been to introduce young researchers to some of the exciting current topics in mathematical physics. These topics often involve analytic techniques that can easily be understood with a dose of physical intuition. In March of 2010, four beautiful lectures were delivered on the campus of the University of Arizona. They included Isoperimetric Inequalities for Eigenvalues of the Laplacian by Rafael Benguria, Universality of Wigner Random Matrices by Laszlo Erdos, Kinetic Theory and the Kac Master Equation by Michael Loss, and Localization in Disordered Media by Gunter Stolz. Additionally, there were talks by other senior scientists and a number of interesting presentations by junior participants. The range of the subjects and the enthusiasm of the young speakers are testimony to the great vitality of this field, and the lecture notes in this volume reflect well the diversity of this school.

In this paper, stationary random processes with fuzzy states are studied. The properties of their numerical characteristics—fuzzy expectations, expectations, and covariance functions—are established. The spectral representation of the covariance function, the generalized Wiener–Khinchin theorem, is proved. The main attention is paid to the problem of transforming a stationary fuzzy random process (signal) by a linear dynamic system. Explicit-form relationships are obtained for the fuzzy expectations (and expectations) of input and output stationary fuzzy random processes. An algorithm is developed and justified to calculate the covariance function of a stationary fuzzy random process at the output of a linear dynamic system from the covariance function of a stationary input fuzzy random process. The results rest on the properties of fuzzy random variables and numerical random processes. Triangular fuzzy random processes are considered as examples.

Let X ( t ) be an ergodic stationary random process or an ergodic homogeneous random field on R m , m ≥ 2 , and let M ( B ) be a mixing homogeneous locally finite random Borel measure with mean density γ on R m , m ≥ 1 . We assume that X and M are independent and possess finite expectations. If { T n } is an increasing sequence of bounded convex sets, containing balls of radii r n → ∞ , then lim n → ∞ 1 λ ( T n ) ∫ T n X ( t ) M ( d t , w ) = γ E [ X ( 0 ) ] a.s. and in L 1 . Special cases are ergodic theorems with averages over finite random sets. Example: If S is an independent-of- X Poisson random set in R m with mean density γ , then lim n → ∞ 1 λ ( T n ) ∑ t ∈ S ∩ T n X ( t ) = γ E [ X ( 0 ) ] a.s. and in L 1 ( card ( S ∩ T n ) < ∞ a.s. ) . These theorems offer a universal way of constructing consistent estimators using observations on finite sets.

We introduce a definition of long range dependence of random processes and fields on an (unbounded) index space $T\\subseteq \\mathbb{R}^d$ in terms of integrability of the covariance of indicators that a random function exceeds any given level. This definition is specifically designed to cover the case of random functions with infinite variance. We show the value of this new definition and its connection to limit theorems via some examples including subordinated Gaussian as well as random volatility fields and time series.

The work is devoted to a problem of the rhythm function estimation of a cyclic random process, which is based on the least squares approximation methods instead of well-known interpolation approach. Analytical dependencies between errors of estimation of a discrete rhythm function and errors of segmentation of a cyclic random process into cycles and zones were constructed. This made it possible to develop a procedure for calculating and controlling errors of estimating rhythm function of a cyclic random process as certain functions of errors of the segmentation method. The general problem of least squares approximation of the rhythm function of a cyclic random process is formulated as a problem of optimal selection of a parametric function derived from a predetermined class of functions that satisfy the necessary and sufficient conditions of the rhythm function of a cyclic random process. New parametric classes of rhythm characteristics of cyclic random processes such as parametric monomials of degree k , parametric logarithmic functions and parametric exponential functions have been built. The advantage of considered method over well-known interpolation approach refers to the improvement of accuracy of rhythm function estimation and reduction of the rhythm function estimation parameters’ number. For example, in presented computer simulation experiment for the parametric class of monomials of degree 2, average value of the mean square errors for 500 simulations in the case of the interpolation is over 40 times higher than the corresponding value for approximation. Moreover, for that parametric class, the number of estimated parameters is almost equal to doubled number of considered cycles in the case of piecewise linear interpolation and is reduced to 1 for least square approximation. The results obtained in the work constitute the basis for improvement of rhythm-adaptive methods and spectral analysis of cyclic random processes, including the area of statistical methods for detecting hidden cyclic structures of the investigated cyclic stochastic signals with an irregular rhythm.

Conservation laws of the generic form c t + f ( c ) x = 0 play a central role in the mathematical description of various engineering related processes. Identification of an unknown flux function f ( c ) from observation data in space and time is challenging due to the fact that the solution c ( x ,  t ) develops discontinuities in finite time. We explore a Bayesian type of method based on representing the unknown flux f ( c ) as a Gaussian random process (parameter vector) combined with an iterative ensemble Kalman filter (EnKF) approach to learn the unknown, nonlinear flux function. As a testing ground, we consider displacement of two fluids in a vertical domain where the nonlinear dynamics is a result of a competition between gravity and viscous forces. This process is described by a multidimensional Navier–Stokes model. Subject to appropriate scaling and simplification constraints, a 1D nonlinear scalar conservation law c t + f ( c ) x = 0 can be derived with an explicit expression for f ( c ) for the volume fraction c ( x ,  t ). We consider small (noisy) observation data sets in terms of time series extracted at a few fixed positions in space. The proposed identification method is explored for a range of different displacement conditions ranging from pure concave to highly non-convex f ( c ). No a priori information about the sought flux function is given except a sound choice of smoothness for the a priori flux ensemble. It is demonstrated that the method possesses a strong ability to identify the unknown flux function. The role played by the choice of initial data c 0 ( x ) as well various types of observation data is highlighted.

By a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. Such random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in Iksanov et al. (2017) and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.