Narrow Results

In this paper we present a panoramic view of numerical simulations associated with nonlinear wave equations which appear in different experimental contexts. Mainly, we deal with scalar wave equations, but also the Maxwell equations in nonlinear media are studied. A basic part of this work is devoted to the construction and verification of numerical schemes on a physical basis. The stochastic perturbations of scalar wave equations are especially analyzed by analytical and numerical approaches. Also, other kinds of perturbations are considered, like nonlocal ones. Finally, a summary of promising experimental results from the numerical simulations of the Maxwell system in a nonlinear media is presented.

We develop systematically a new unifying approach to the analysis of linear stochastic, quantum stochastic and even deterministic equations in Banach spaces. Solutions to a wide class of these equations (in particular those describing the processes of continuous quantum measurements) are proved to coincide with the interaction representations of the solutions to certain Dirac type equations with boundary conditions in pseudo-Fock spaces. The latter are presented as the semiclassical limit of an appropriately dressed unitary evolutions corresponding to a boundary-value problem for rather general Schrödinger equations with bounded below Hamiltonians.

In this paper, we study a stochastic system of differential equations with nonlocal discrete diffusion. For two types of noises, we study the existence of either positive or probability solutions. Also, we analyze the asymptotic behavior of solutions in the long term, showing that under suitable assumptions they tend to a neighborhood of the unique deterministic fixed point. Finally, we perform numerical simulations and discuss the application of the results to life tables for mortality in Spain.

In this paper, we deal with the approximate controllability of fractional stochastic delay differential inclusions of order $r\in (1,2)$. By using fractional calculus, stochastic analysis, the theory of cosine family and Dhage fixed point techniques, a new set of necessary and sufficient conditions are formulated which guarantees the approximate controllability of the nonlinear fractional stochastic system. In particular, the results are established with the assumption that the associated linear part of the system is approximately controllable. Further, the result is extended to obtain the conditions for the solvability of controllability results for fractional inclusions with nonlocal conditions. Finally, an example is presented to illustrate the theory of the obtained result.

Stochastic Einstein equations are considered when three-dimensional space metric γ_ij are stochastic functions. The probability density for the stochastic quantities is connected with Perelman's entropy functional. As an example, the Friedman Universe is considered. It is shown that for the Friedman Universe the dynamical evolution is not changed. The connection between general relativity and Ricci flow is discussed.

I discuss the connection of the three different questions: The existence of the Gibbs steady state distributions for the stochastic differential equations, the notion and the existence of the conservation laws for such equations, and the convergence of the smooth random perturbations of dynamical systems to stochastic differential equations in the Ito sense. I show that in all cases one needs to include some additional term in the standard form of stochastic equation. I call such approach to describing the influence of the noise on the dynamical systems the "stochastic deformation" to distinguish it from the conventional "stochastic perturbation". I also discuss some consequences of this approach, in particular, a connection between the noise intensity and the temperature. This connection is known in physics (for the case of linear system of differential equations) as "fluctuation-dissipation theorem" (L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 9, Statistical Physics. Part 2). In conclusion, I present an interesting physical example of the dynamics of magnetic dipole in a random magnetic field.

The aim of this paper is to discuss some problems connected with reconstructibility of trajectories of a class of stochastic equations. In particular we address Cauchy-like problems for the so-called generalized birth-and-death processes. This type of processes are a simple, natural formal framework for modelling a vast variety of biological processes such as population dynamics, genome evolution and somatic evolution of cancers. We describe how empirical data, e.g., mean values of some attributes of systems in questions, can be used to find trajectories of these systems.