Stochastics and Dynamics

In this paper, we investigate the stochastic Cahn–Hilliard–Navier–Stokes equations in two-dimensional spaces. Applying the Maslowski–Seidler method, we establish the existence of invariant measure in state space $L_x^2\times H^1$ with the weak topology. We also prove the existence of global pathwise solutions using the stochastic compactness argument.

Although chaotic attractors for autonomous dynamical systems show sensitive dependence on initial conditions, they also typically support a physical or natural measure that characterizes the statistical behavior of almost all initial conditions near the attractor with respect to a background measure such as Lebesgue. In this paper, we identify conditions under which a nonautonomous system that limits as $t\to -\infty$ to an autonomous system with a physical measure is guaranteed to possess a “nonautonomous physical measure” that limits to the physical measure of the autonomous system.

In this paper, we consider the asymptotic behavior of solutions for nonautonomous stochastic parabolic equation with nonstandard growth condition driven by nonlinear multiplicative noise for the first time. First, by making use of variational method, we prove the existence and uniqueness of solutions, and then the mean random dynamical systems generated by stochastic parabolic equations with variable exponents are obtained. Finally, due to the influence of variable indexes (dependent on space variable), we show the existence of weak mean random attractors under suitable assumptions on the variable exponents and the diffusion term.

In this paper, we introduce fuzzy stochastic differential equations (FSDEs) driven by sub-fractional Brownian motion (SFBM) which are applied to describe phenomena subjected to randomness and fuzziness simultaneously. The SFBM is an extension of the Brownian motion that retains many properties of fractional Brownian motion (FBM), but not the stationary increments. This property makes SFBM a possible candidate for models that include long-range dependence, self-similarity, and non-stationary increments which is suitable for the construction of stochastic models in finance and non-stationary queueing systems. We apply an approximation method to stochastic integrals, and a decomposition of the SFBM to find the existence and uniqueness of the solutions.

In this paper, we study the asymptotic behavior of maximum likelihood estimators for Ornstein–Uhlenbeck process with large linear drift $d{X}_t=-\frac{1}{𝜀}(𝜃X_t-{𝜀}^{\frac{1}{2}}\nu )dt+d{B}_t$, $0\le t\le T$, where $𝜃,\nu \in ℝ$, and ${\{B_t\}}_{t\ge 0}$ is a given standard Brownian motion. The law of iterated logarithm, consistency and asymptotic distributions of the estimators are discussed based on the continuous observation ${\{X_t\}}_{t\in [0,T]}$ as $𝜀\to 0$.

We obtain large and moderate deviation estimates, as well as concentration inequalities, for a class of nonuniformly expanding maps with stretched exponential decay of correlations. In the large deviation regime, we also exhibit examples showing that the obtained upper bounds are essentially optimal.

Various mixing properties of $\beta$-, ${\beta }^{\prime }$- and Gaussian-Delaunay tessellations in ${ℝ}^{d-1}$ are studied. It is shown that these tessellation models are absolutely regular, or $\beta$-mixing. In the $\beta$- and the Gaussian case exponential bounds for the absolute regularity coefficients are found. In the ${\beta }^{\prime }$-case these coefficients show a polynomial decay only. In the background are new and strong concentration bounds on the radius of stabilization of the underlying construction. Using a general device for absolutely regular stationary random tessellations, central limit theorems for a number of geometric parameters of $\beta$- and Gaussian-Delaunay tessellations are established. This includes the number of $k$-dimensional faces and the $k$-volume of the $k$-skeleton for $k\in \{0,1,\dots ,d-1\}$.

We consider the problem of determining two-sided exit probabilities for a compound renewal process with drift and two-sided jumps. In certain cases the problem can be reduced to determining the distribution of a random sum of i.i.d. random variables. In a general situation this problem is reduced to solving a certain integral equation. We obtain explicit expressions for the escape probability for several classes of jump times and sizes.

Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy the canonical anti-commutation relation (CAR) in equal time. This paper aimed to discuss the classical reduction and ergodicity of quantum exclusion semigroups constructed by QBN. We first study the classical reduction of the quantum semigroups to an Abelian algebra of diagonal elements and the space of off-diagonal elements. We then provide an explicit representation formula by separating the action on off-diagonal and diagonal operators, on which they are reduced to the semigroups of classical Markov chains. Finally, we prove that the asymptotic behavior of the quantum semigroups is equivalent to one of its associated Markov chains, and that the semigroups restricted to the off diagonal space of operators have a zero limit.