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Nonstandard ergodic averages can be defined for a measure-preserving action of a group on a probability space, as a natural extension of classical (nonstandard) ergodic averages. We extend the one-dimensional theory, obtaining L^1 pointwise ergodic theorems for several kinds of nonstandard sparse group averages, with a special focus on the group Z^d. Namely, we extend results for sparse block averages and sparse random averages to their analogues on virtually nilpotent groups, and extend Christs result for sparse deterministic sequences to its analogue on Z^d. The second and third results have two nontrivial variants on Z^d: a native d-dimensional average and a product average from the 1-dimensional averages.
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In contrast to existing works on stochastic averaging on finite intervals, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for semilinear stochastic ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost periodic, almost automorphic etc) coefficients. Under some appropriate conditions we prove that there exists a unique recurrent solution to the original equation, which possesses the same recurrence property as the coefficients, in a small neighborhood of the stationary solution to the averaged equation, and this recurrent solution converges to the stationary solution of averaged equation uniformly on the whole real axis when the time scale approaches zero.
Let $(X,mathscr{B}, mu,T,d)$ be a measure-preserving dynamical system with exponentially mixing property, and let $mu$ be an Ahlfors $s$-regular probability measure. The dynamical covering problem concerns the set $E(x)$ of points which are covered by the orbits of $xin X$ infinitely many times. We prove that the Hausdorff dimension of the intersection of $E(x)$ and any regular fractal $G$ equals $dim_{rm H}G+alpha-s$, where $alpha=dim_{rm H}E(x)$ $mu$--a.e. Moreover, we obtain the packing dimension of $E(x)cap G$ and an estimate for $dim_{rm H}(E(x)cap G)$ for any analytic set $G$.
In this paper, we establish the second Bogolyubov theorem and global averaging principle for stochastic partial differential equations (in short, SPDEs) with monotone coefficients. Firstly, we prove that there exists a unique $L^{2}$-bounded solution to SPDEs with monotone coefficients and this bounded solution is globally asymptotically stable in square-mean sense. Then we show that the $L^{2}$-bounded solution possesses the same recurrent properties (e.g. periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, Levitan almost periodic, etc.) in distribution sense as the coefficients. Thirdly, we prove that the recurrent solution of the original equation converges to the stationary solution of averaged equation under the compact-open topology as the time scale goes to zero--in other words, there exists a unique recurrent solution to the original equation in a neighborhood of the stationary solution of averaged equation when the time scale is small. Finally, we establish the global averaging principle in weak sense, i.e. we show that the attractor of original system tends to that of the averaged equation in probability measure space as the time scale goes to zero. For illustration of our results, we give two applications, including stochastic reaction diffusion equations and stochastic generalized porous media equations.
We study reduction schemes for functions of many variables into system of functions in one variable. Our setting includes infinite-dimensions. Following Cybenko-Kolmogorov, the outline for our results is as follows: We present explicit reductions schemes for multivariable problems, covering both a finite, and an infinite, number of variables. Starting with functions in many variables, we offer constructive reductions into superposition, with component terms, that make use of only functions in one variable, and specified choices of coordinate directions. Our proofs are transform based, using explicit transforms, Fourier and Radon; as well as multivariable Shannon interpolation.
We present the modified approach to the classical Bogolyubov-Krylov averaging, developed recently for the purpose of PDEs. It allows to treat Lipschitz perturbations of linear systems with pure imaginary spectrum and may be generalized to treat PDEs with small nonlinearities.
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