Limit Gaussian Behavior of the Solutions of the Multidimensional Burger'; Equation with Weak-Dependent Initial Conditions

Asymptotic normality of the solutions of the initial-value problems for multidimensional Burgers' equation with weakly dependent possible non-Gaussian initial condition is proved.     

Gaussian Limiting Behavior of the Rescaled Solution to the Linear Korteweg–de Vries Equation with Random Initial Conditions

We analyze the asymptotic behavior of the rescaled solution to the linear Korteweg–de Vries equation when the initial conditions are supposed to be random and weakly dependent. By means of the method of moments we prove the Gaussianity of the limiting process and we present its correlation function. The same technique is applied to the analysis of another third-order heat-type equation.     

Spherical level-crossing measures for chi-square random fields

Spherical measures of exceeding a low level are considered for strongly dependent chi-square fields. The limit distributions are sums of multiple stochastic integrals.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 118–128, 1989.     

Central limit theorem for nonlinear transforms of gaussian random vector fields

We prove a central limit theorem for integral functionals of nonlinear transforms of homogeneous, isotropic Gaussian random vector fields.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1057–1063, August, 1990.     

Deformation of a three-layer elastoplastic beam on an elastic foundation

We study the thermal force bending of an elastoplastic three-layer beam with a rigid filler; the beam is connected with an elastic foundation. The broken normal hypothesis is adopted to describe the kinematics of a packet nonsymmetric with respect to the thickness. The foundation reaction is described by Winkler’s model. The system of equilibrium equations and its exact solution in displacements are obtained and numerical results for a three-layer metal-polymeric beam are presented.     

Scaling Limits of Solutions of the Heat Equation for Singular Non-Gaussian Data

Limiting distributions of the parabolically rescaled solutions of the heat equation with singular non-Gaussian initial data with long-range dependence are described in terms of their multiple stochastic integral representations.     

Scaling Laws for the Multidimensional Burgers Equation with Quadratic External Potential

The reordering of the multidimensional exponential quadratic operator in coordinate-momentum space (see X. Wang, C.H. Oh and L.C. Kwek (1998). J. Phys. A.: Math. Gen. 31:4329–4336) is applied to derive an explicit formulation of the solution to the multidimensional heat equation with quadratic external potential and random initial conditions. The solution to the multidimensional Burgers equation with quadratic external potential under Gaussian strongly dependent scenarios is also obtained via the Hopf-Cole transformation. The limiting distributions of scaling solutions to the multidimensional heat and Burgers equations with quadratic external potential are then obtained under such scenarios. AMS Subject Classifications: 60G60, 60G15, 62M15, 60H15     

Spectral Properties of Uperpositions of Ornstein-Uhlenbeck Type Processes

Stationary processes with prescribed one-dimensional marginal laws and long-range dependence are constructed. The asymptotic properties of the spectral densities are studied. The possibility of Mittag-Leffler decay in the autocorrelation function of superpositions of Ornstein-Uhlenbeck type processes is proved. AMS 2000 Subject Classification Primary 60E07, 60G10, 60G18 Secondary 62M10, 62P05 Supported by the Danish National Research Foundation and EPSRC grant RCMT091. Partially supported by the ARC grants DP 0345577, DP0559807 and EPSRC grant RCMT091.