arXiv subjects
Publications and source records attributed to Jianliang Zhai.
In this paper we prove uniform Hausdorff and packing dimension results for the images of a large family of Markov processes. The main tools are the two covering principles of Xiao (second author). As applications, uniform Hausdorff and packing dimension results for certain classes of Lévy processes, stable jump diffusion and non-symmetric stable-like processes are obtained.
In this paper, we establish a large deviation principle for stochastic models of two-dimensional second grade fluids driven by Lévy noise. The weak convergence method introduced by Budhiraja, Dupuis and Maroulas in [5] plays a key role.
In this paper, we establish the existence and uniqueness of both mild(/variational) solutions and weak (in the sense of PDE) solutions of coupled system of 2D stochastic Chemotaxis-Navier-Stokes equations. The mild/variational solution is obtained through a fixed point argument in a purposely constructed Banach space. To get the weak solution we first prove the existence of a martingale weak solution and then we show that the pathwise uniqueness holds for the martingale solution.
We consider a stochastic model of incompressible non-Newtonian fluids of second grade on a bounded domain of $\mathbb{R}^2$ driven by Lévy noise. Applying the variational approach, global existence and uniqueness of strong probabilistic solution is established.
For a heat equation with memory driven by a Lévy-type noise we establish the existence of a unique solution. The main part of the article focuses on the Freidlin-Wentzell large deviation principle of the solutions of heat equation with memory driven by a Lévy-type noise. For this purpose, we exploit the recently introduced weak convergence approach.
The limiting behavior of stochastic evolution processes with small noise intensity $ε$ is investigated in distribution-based approach. Let $μ^ε$ be stationary measure for stochastic process $X^ε$ with small $ε$ and $X^{0}$ be a semiflow on a Polish space. Assume that $\{μ^ε: 0<ε\leqε_0\}$ is tight. Then all their limits in weak sense are $X^0-$invariant and their supports are contained in Birkhoff center of $X^0$. Applications are made to various stochastic evolution systems, including stochastic ordinary differential equations, stochastic partial differential equations, stochastic functional differential equations driven by Brownian motion or Lévy process.
In this paper, we prove a central limit theorem and estabilish a moderate deviation principle for stochastic models of incompressible second fluids. The weak convergence method inreoduced by [4] plays an important role.
In this paper, we establish the large deviation principle for 3D stochastic primitive equations with small perturbation multiplicative noise. The proof is mainly based on the weak convergence approach.
In this paper, we establish a large deviation principle for a type of stochastic partial differential equations (SPDEs) with locally monotone coefficients driven by Lévy noise. The weak convergence method plays an important role.
Motivated by the work of Busse et al. [6] on turbulent convection in a rotating layer, we exploit the long-run behavior for stochastic Lotka-Volterra (LV) systems both in pull-back trajectory and in stationary measure. It is proved stochastic decomposition formula describing the relation between solutions of stochastic and deterministic LV systems and stochastic Logistic equation. By virtue of this formula, it is verified that every pull-back omega limit set is an omega limit set for deterministic LV systems multiplied by the random equilibrium of the stochastic Logistic equation. This formula is used to derive the existence of a stationary measure, its support and ergodicity. We prove the tightness for the set of stationary measures and the invariance for their weak limits as the noise intensity vanishes, whose supports are contained in the Birkhoff center. The developed theory is successfully utilized to completely classify three dimensional competitive stochastic LV systems into $37$ classes. Time average probability measures weakly converge to an ergodic stationary measure on the attracting domain of an omega limit set in all classes except class 27 c). Among them there are two classes possessing a continuum of random closed orbits and ergodic stationary measures supported in cone surfaces, which weakly converge to the Haar measures of periodic orbits as the noise intensity vanishes. In the exceptional class, almost every pull-back trajectory cyclically oscillates around the boundary of the stochastic carrying simplex characterized by three unstable stationary solutions. The limit for the time average probability measures is neither unique nor ergodic. These are subject to turbulent characteristics.
In this paper, we prove a central limit theorem and establish a moderate deviation principle for 2D stochastic hydrodynamical type systems with multiplicative noise in unbounded domains, which covers 2D Navier-Stokes equations, 2D MHD models and the 2D magnetic B?enard problem and also shell models of turbulence. The weak convergence method plays an important role in obtaining the moderate deviation principle.
In this paper, we established a large deviation principle for stochastic models of incompressible second grade fluids. The weak convergence method introduced by \cite{Budhiraja-Dupuis} plays an important role.
In this paper, we establish a moderate deviation principle for two-dimensional stochastic Navier-Stokes equations driven by multiplicative $L\acute{e}vy$ noises. The weak convergence method introduced by Budhiraja, Dupuis and Ganguly in arXiv:1401.73v1 plays a key role.
We study SPDEs with two reflecting walls $Λ^1$, $Λ^2$ and two singular drifts $\frac{c_1}{(X-Λ^1)^{\vartheta}}$, $\frac{c_2}{(Λ^2-X)^{\vartheta}}$, driven by space-time white noise. First, we establish the existence and uniqueness of the solutions $X$ for $\vartheta\geq 0$. Second, we obtain the following pathwise properties of the solutions $X$. If $\vartheta>3$, then a.s. $Λ^1<X<Λ^2$ for all $t\geq0$; If $0<\vartheta<3$, then $X$ hits $Λ^1$ or $Λ^2$ with positive probability in finite time. Thus $\vartheta=3$ is the critical parameter for $X$ to hit reflecting walls.
In this paper, we establish the exponential mixing property of stochastic models for the incompressible second grade fluid. The general criterion established by Cyril Odasso plays an important role.
In this paper, we establish a large deviation principle for a fully non-linear stochastic evolution equation driven by both Brownian motions and Poisson random measures on a given Hilbert space $H$. The weak convergence method plays an important role.