A Statistical Theory of Linear Dissipative Systems

Stability of Quasi-linear Hyperbolic Dissipative Systems

1. Introduction In this work we want to explore the relationship between certain eigenvalue condition for the symbols of first order partial differential operators describing evolution processes and the linear and nonlinear stability of their stationary solutions. Consider the initial value problem for the following general first order quasi-linear system of equations

Deformation quantization of linear dissipative systems

A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of a non-stationary Poisson bracket, i.e. the corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation we develop a consistent deformation quantization pro...

Statistical Theory of Non-linear Stochastic Systems I

Upper Semi-continuity of Stationary Statistical Properties of Dissipative Systems

We show that stationary statistical properties for uniformly dissipative dynamical systems are upper semi-continuous under regular perturbation and a special type of singular perturbation in time of relaxation type. The results presented are applicable to many physical systems such as the singular limit of infinite Prandtl-Darcy number in the Darcy-Boussinesq system for convection in porous med...

Numerical algorithms for stationary statistical properties of dissipative dynamical systems

It is well-known that physical laws for large chaotic dynamical systems are revealed statistically. The main concern of this manuscript is numerical methods for dissipative chaotic infinite dimensional dynamical systems that are able to capture the stationary statistical properties of the underlying dynamical systems. We first survey results on temporal and spatial approximations that enjoy the...