Stochastics and Dynamics最新文献

The reflection problem on the positive half-line with reflection at zero for a time-dependent stochastic differential equations driven by standard and fractional Brownian motion with Hurst parameter $H>\frac{1}{2}$ is considered. We prove the existence of weak solutions by using Euler scheme. Moreover, we show that pathwise uniqueness holds and a strong solution exists in the case of additive fractional noise and also up to a stopping time $\tau$ for the multiplicative case, but remains an open question beyond $\tau$.

In a vast area of probabilistic limit theorems for dynamical systems with chaotic behaviors always only functional form (exponential, power, etc.) of the asymptotic laws and of convergence rates were studied. However, for basically all applications, e.g., for computer simulations, development of algorithms to study chaotic dynamical systems numerically, as well as for design and analysis of real (e.g., in physics) experiments, the exact values (or at least estimates) of constants (parameters) of the functions, which appear in the asymptotic laws and rates of convergence, are of primary interest. In this paper, we provide such estimates of constants (parameters) in the central limit theorem, large deviations principle, law of large numbers and the rate of correlations decay for strongly chaotic dynamical systems.

Consider a non-autonomous 2D-Ginzburg–Landau equation driven by Wong–Zakai noise or white noise, respectively, we first show the existence of pullback random attractors, which are random compact attracting sets indexed by two parameters: the size of Wong–Zakai noise and the current time. We then establish the robustness of the attractors when both parameters are simultaneously convergent. An essential difficulty arises from the possible loss of the convergence of solutions and only part convergence of solutions is available, which is a new phenomenon for 2D-GL equation distinguishing with the 1D case. So, by using part joint-convergence, regularity, eventual local-compactness and recurrence, we establish a binary robustness theorem of pullback random attractors and apply it to the weakly dissipative stochastic equation.

In this paper, the multiscale stochastic 3D generalized Navier–Stokes equations are studied. By using Khasminkii’s time discretization approach and the technique of stopping time, the strong averaging principle for stochastic 3D generalized Navier–Stokes equations is proved in the space ${ℍ}^1({𝕋}^3)$.

In this paper, we consider the reflected backward stochastic differential equations driven by $G$-Brownian motion (reflected $G$-BSDEs) with time-varying Lipschitz coefficients. We obtain the uniqueness result by establishing a priori estimates. For the existence, the solution can be approximated by a family of reflected $G$-BSDEs with Lipschitz conditions and by penalized $G$-BSDEs with time-varying coefficients. The latter approximation is useful to get the comparison theorem. Finally, we study the reflected $G$-BSDEs with infinite time horizon.

This paper is concerned with the asymptotic behaviors of the solutions of an impulsive stochastic neural field lattice model driven by nonlinear noise. We first show the existence and uniqueness of weak pullback mean random attractors for the impulsive stochastic systems. Then by the properties of Markov processes, the existence of evolution system of measures for the impulsive stochastic systems is established. To this end, we employ the idea of uniform estimates on the tails of the solutions to show the tightness of a family of distributions of the solutions of the lattice systems.