Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations

Planar random walk in a stratified quasi-periodic environment

We study the recurrence of inhomogeneous Markov chains in the plane, when the environment is horizontally stratified and the heterogeneity of quasi-periodic type.

Periodic, Quasi-Periodic, Almost Periodic, Almost Automorphic, Birkhoff Recurrent and Poisson Stable Solutions for Stochastic Differential Equations

The paper is dedicated to studying the problem of Poisson stability (in particular stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, Bohr almost automorphy, Birkhoff recurrence, almost recurrence in the sense of Bebutov, Levitan almost periodicity, pseudo-periodicity, pseudo-recurrence, Poisson stability) of solutions for semi-linear stochastic equation $$ dx(t)=(Ax(t)+f(t,x(t)))dt +g(t,x(t))dW(t)\quad (*) $$ with exponentially stable linear operator $A$ and Poisson stable in time coefficients $f$ and $g$. We prove that if the functions $f$ and $g$ are appropriately "small…

Stochastic Bifurcation of Pathwise Random Almost Periodic and Almost Automorphic Solutions for Random Dynamical Systems

In this paper, we introduce concepts of pathwise random almost periodic and almost automorphic solutions for dynamical systems generated by non-autonomous stochastic equations. These solutions are pathwise stochastic analogues of deterministic dynamical systems. The existence and bifurcation of random periodic (random almost periodic, random almost automorphic) solutions have been established for a one-dimensional stochastic equation with multiplicative noise.

次椭圆情形下拟周期强迫二维随机Navier-Stokes方程的指数混合与极限定理

Exponential mixing and limit theorems of quasi-periodically forced 2D stochastic Navier-Stokes Equations in the hypoelliptic setting

该研究针对拟周期与退化噪声驱动的二维随机Navier-Stokes方程，证明了存在指数吸引所有解分布的拟周期不变测度，并建立了带显式收敛速率的强大数定律与中心极限定理，揭示了收敛速率与拟周期频率丢番图性质的关联。

Almost sure recovery in quasi-periodic structures

We study random perturbations of quasi-periodic uniformly discrete sets in the $d$-dimensional euclidean space. By means of Diffraction Theory, we find conditions under which a quasi-periodic set $X$ can be almost surely recovered from its random perturbations. This extends the recent periodic case result of Yakir from "Recovery the lattice from its random perturbations" IMRN 2020 (arXiv:200201508).

Quasi-periodic and almost periodic homogenizations of integro-differential equations with Levy operators

In this paper, we studied quasi-periodic and almost periodic homogenizations of integro-differential equations with Levy operators, which contain alpha stable Levy densities. This is the intermediate stage between the periodic homogenization and the stochastic homogenization of the non-local problem, and is the straight forward generalization of the case of partial differential equations.