Global properties of invariant measures

We study global regularity properties of invariant measures associated with second order differential operators in R^N. Under suitable conditions, we prove global boundedness of the density, Sobolev regularity, a Harnack inequality and pointwise upper and lower bounds.     

Dissipativity and invariant measures for stochastic Navier-Stokes equations

A 2-dimensional stochastic Navier-Stokes equation with a general white noise is considered. The aim is to prove the existence of invariant measures, using a new dissipativity property of the stochastic dynamic.     

Existence of invariant measures of stochastic systems with delay in the highest order partial derivatives

In this note, we shall consider the existence of invariant measures for a class of infinite dimensional stochastic functional differential equations with delay whose driving semigroup is eventually norm continuous. The results obtained are applied to stochastic heat equations with distributed delays which appear in such terms having the highest order partial derivatives. In these systems, the associated driving semigroups are generally non eventually compact.     

Finite-dimensionality of bounded invariant sets for Navier-stokes systems and other dissipative systems

One proves the finite-dimensionality of a bounded set M of a Hilbert space H, negatively invariant relative to a transformation V, possessing the following properties: For any points υ and \(\tilde \upsilon\) of the set M one has $$\left\| {V(\upsilon ) - V(\tilde \upsilon )} \right\| \leqslant \ell \left\| {\upsilon - \tilde \upsilon } \right\|$$ , while $$\left\| {Q_n V(\upsilon ) - Q_n V(\tilde \upsilon )} \right\| \leqslant \delta \left\| {\upsilon - \tilde \upsilon } \right\|, \delta< 1$$ ,δ<1 where Q_n is the orthoprojection onto a subspace of codimension n. With the aid of this result and of the results found in O. A. Ladyzhenskaya's paper “On the dynamical system generated by the Navier-stokes equations” (J. Sov. Math.,3, No. 4 (1975)) one establishes the finite-dimensionality of the complete attractor for two-dimensional Navier-Stokes equations. The same holds for many other dissipative problems.     

Convergence of invariant measures for singular stochastic diffusion equations

It is proved that the solutions to the singular stochastic p$p$-Laplace equation, p∈(1,2)$p\in (1,2)$ and the solutions to the stochastic fast diffusion equation with nonlinearity parameter r∈(0,1)$r\in (0,1)$ on a bounded open domain Λ⊂R^d$\Lambda \subset {\mathbb{R}}^d$ with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters p$p$ and r$r$ respectively (in the Hilbert spaces L²(Λ)$L^2\left(\Lambda \right)$, H⁻¹(Λ)$H^{-1}\left(\Lambda \right)$ respectively). The highly singular limit case p=1$p=1$ is treated with the help of stochastic evolution variational inequalities, where P$\mathbb{P}$-a.s. convergence, uniformly in time, is established.     

Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems

This paper first summarizes the theory of quasi-periodic bifurcations for dissipative dynamical systems. Then it presents algorithms for the computation and continuation of invariant circles and of their bifurcations. Finally several applications are given for quasiperiodic bifurcations of Hopf, saddle-node and period-doubling type.     

Dynamics of stochastic partly dissipative systems on time-varying domain

This article is devoted to stochastic partly dissipative systems (SPDS) defined on time-varying domain expending in time. The definition of a space–time trace class Wiener process on time-varying domain is presented by combining a time trace class Wiener process with a spatial intensity defined on time-varying domain. Some nice properties for Wiener process on time-varying domain are also presented. We develop a new penalty method to establish the existence and uniqueness of variational solution of SPDS with additive noise on time-varying domain. The existence of global attractor for the process generated by variational solution is also obtained.     

Linear invariant measures for recurrent linear systems

We consider a self-adjoint operator defined by a bidimensional linear system. We extend the Ishii-Pastur-Kotani theory that allows us to identify the absolutely continuous spectrum. From here we deduce that for almost everyE with null Lyapunov exponent the real projective flow admits absolutely continuous invariant measures with square integrable density function.     

On invariant tori of stochastic systems on a plane

We study invariant tori of stochastic systems of the ltd type on a plane and present conditions for stability of such sets in probability.     

Some symmetric systems of minimal surfaces

In this paper we use a method of an earlier paper in order to prove the existence of some symmetric systems of minimal surfaces bounded by curves having self-intersections.     

Finite-dimensional distributions of invariant measure in nonlinear stochastic differential systems

The article studies stochastic processes defined by Ito stochastic differential equations with Wiener white noise. Methods are considered that find exact expressions for finite-dimensional densities of random stationary and nonstationary (in the narrow sense) processes based on construction of integral invariants of specially chosen systems of ordinary differential equations. Translated from Algoritmy Upravleniya i Identifikatsii, pp. 129–140, 1997.     

Generic invariant measures for iterated systems of interval homeomorphisms

Archiv der Mathematik - It is well known that iterated function systems generated by orientation preserving homeomorphisms of the unit interval with positive Lyapunov exponents at its ends admit a...