arXiv subjects

Jinqiao Duan

Publications and source records attributed to Jinqiao Duan.

At least 217 records · Page 12Linked to original sources

arXiv2011

Global Well-posedness of the Stochastic Kuramoto-Sivashinsky Equation with Multiplicative Noise

Global well-posedness of the initial-boundary value problem for the stochastic Kuramoto-Sivashinsky equation in a bounded domain $D$ with a multiplicative noise is studied. It is shown that under suitable sufficient conditions, for any initial data $u_0\in L^2(D\times Ω)$ this problem has a unique global solution $u$ in the space $L^2(Ω,C([0,T],L^2({D})))$ for any $T>0$, and the solution map $u_0\mapsto u$ is Lipschitz continuous.

math.AP↗

arXiv2010

Effective dynamicsof a coupled microscopic-macroscopic stochastic system

A conceptual model for microscopic-macroscopic slow-fast stochastic systems is considered. A dynamical reduction procedure is presented in order to extract effective dynamics for this kind of systems. Under appropriate assumptions, the effective system is shown to approximate the original system, in the sense of a probabilistic convergence.

math.PR↗

arXiv2010

Evolution systems of measures for stochastic flows

A new concept of {\em an evolution system of measures for stochastic flows} is considered. It corresponds to the notion of an invariant measure for random dynamical systems (or cocycles). The existence of evolution systems of measures for asymptotically compact stochastic flows is obtained. For a white noise stochastic flow, there exists a one to one correspondence between evolution systems of measures for a stochastic flow \emph{and} evolution systems of measures for the associated Markov transition semigroup. As an application, an alternative approach for evolution systems of measures of 2D stochastic Navier-Stokes equations with a time-periodic forcing term is presented.

math.DS↗

arXiv2010

Geometric shape of invariant manifolds for a class of stochastic partial differential equations

Invariant manifolds play an important role in the study of the qualitative dynamical behaviors for nonlinear stochastic partial differential equations. However, the geometric shape of these manifolds is largely unclear. The purpose of the present paper is to try to describe the geometric shape of invariant manifolds for a class of stochastic partial differential equations with multiplicative white noises. The local geometric shape of invariant manifolds is approximated, which holds with significant likelihood. Furthermore, the result is compared with that for the corresponding deterministic partial differential equations.

math.AP↗

arXiv2010

Quantifying Uncertainties in Complex Systems

Uncertainties are abundant in complex systems. Mathematical models for these systems thus contain random effects or noises. The models are often in the form of stochastic differential equations, with some parameters to be determined by observations. The stochastic differential equations may be driven by Brownian motion, fractional Brownian motion or Lévy motion. After a brief overview of recent advances in estimating parameters in stochastic differential equations, various numerical algorithms for computing parameters are implemented. The numerical simulation results are shown to be consistent with theoretical analysis. Moreover, for fractional Brownian motion and $α-$stable Lévy motion, several algorithms are reviewed and implemented to numerically estimate the Hurst parameter $H$ and characteristic exponent $α$.

math.NA↗

arXiv2010

Large deviations for slow-fast stochastic partial differential equations

A large deviation principle is derived for stochastic partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.

math.PR↗

arXiv2009

A sufficient condition for bifurcation in random dynamical systems

Some properties of random Conley index are obtained and then a sufficient condition for the existence of abstract bifurcation points for both discrete-time and continuous-time random dynamical systems is presented. This stochastic bifurcation phenomenon is demonstrated by a few examples.

math.DS↗

arXiv2009

Canonical Sample Spaces for Random Dynamical Systems

This is an overview about natural sample spaces for differential equations driven by various noises. Appropriate sample spaces are needed in order to facilitate a random dynamical systems approach for stochastic differential equations. The noise could be white or colored, Gaussian or non-Gaussian, Markov or non-Markov, and semimartingale or non-semimartingale. Typical noises are defined in terms of Brownian motion, Levy motion and fractional Brownian motion. In each of these cases, a canonical sample space with an appropriate metric (or topology that gives convergence concept) is introduced. Basic properties of canonical sample spaces, such as separability and completeness, are then discussed. Moreover, a flow defined by shifts, is introduced on these canonical sample spaces. This flow has an invariant measure which is the probability distribution for Brownian motion, or Levy motion or fractional Brownian motion. Thus canonical sample spaces are much richer in mathematical structures than the usual sample spaces in probability theory, as they have metric or topological structures, together with a shift flow (or driving flow) defined on it. This facilitates dynamical systems approaches for studying stochastic differential equations.

math.DS↗

arXiv2009

Pseudorandom Numbers for Conformal Measure

We propose a new algorithm for generating pseudorandom (pseudo-generic) numbers of conformal measures of a continuous map T acting on a compact space X and for a Holder continuous potential F. In particular, we show that this algorithm provides good approximations to generic points for hyperbolic rational functions of degree two and the potential -h log|T'|, where h denotes the Hausdorff dimension of the Julia set of T .

math.DS↗

arXiv2009

Random Chain Recurrent Sets for Random Dynamical Systems

It is known by the Conley's theorem that the chain recurrent set $CR(ϕ)$ of a deterministic flow $ϕ$ on a compact metric space is the complement of the union of sets $B(A)-A$, where $A$ varies over the collection of attractors and $B(A)$ is the basin of attraction of $A$. It has recently been shown that a similar decomposition result holds for random dynamical systems on noncompact separable complete metric spaces, but under a so-called \emph{absorbing condition}. In the present paper, the authors introduce a notion of random chain recurrent sets for random dynamical systems, and then prove the random Conley's theorem on noncompact separable complete metric spaces \emph{without} the absorbing condition.

math.DS↗

arXiv2009

Predictability in Spatially Extended Systems with Model Uncertainty

Macroscopic models for spatially extended systems under random influences are often described by stochastic partial differential equations (SPDEs). Some techniques for understanding solutions of such equations, such as estimating correlations, Liapunov exponents and impact of noises, are discussed. They are relevant for understanding predictability in spatially extended systems with model uncertainty, for example, in physics, geophysics and biological sciences. The presentation is for a wide audience.

math.DS↗

arXiv2009

State space decomposition for nonautonomous dynamical systems

Decomposition of state spaces into dynamically different components is helpful for the understanding of dynamical behaviors of complex systems. A Conley type decomposition theorem is proved for nonautonomous dynamical systems defined on a non-compact but separable state space. Namely, the state space can be decomposed into a chain recurrent part and a gradient-like part. This result applies to both nonautonomous ordinary differential equations on Euclidean space (which is only locally compact), and nonautonomous partial differential equations on infinite dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier-Stokes system, under time-dependent forcing.

math.DS↗

arXiv2009

A Stochastic Approach for Parameterizing Unresolved Scales in a System with Memory

Complex systems display variability over a broad range of spatial and temporal scales. Some scales are unresolved due to computational limitations. The impact of these unresolved scales on the resolved scales needs to be parameterized or taken into account. One stochastic parameterization scheme is devised to take the effects of unresolved scales into account, in the context of solving a nonlinear partial differential equation with memory (a time-integral term), via large eddy simulations. The obtained large eddy simulation model is a stochastic partial differential equation. Numerical experiments are performed to compare the solutions of the original system and of the stochastic large eddy simulation model.

math.NA↗

arXiv2009

Synchronization of dissipative dynamical systems driven by non-Gaussian Levy noises

Dynamical systems driven by Gaussian noises have been considered extensively in modeling, simulation and theory. However, complex systems in engineering and science are often subject to non-Gaussian fluctuations or uncertainties. A coupled dynamical system under non- Gaussian Levy noises is considered. After discussing cocycle prop- erty, stationary orbits and random attractors, a synchronization phe- nomenon is shown to occur, when the drift terms of the coupled system satisfy certain dissipativity and integrability conditions. The synchro- nization result implies that coupled dynamical systems share a dy- namical feature in some asymptotic sense.

math.DS↗

arXiv2009

Invariant manifolds for random and stochastic partial differential equations

Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable and pseudo-unstable manifolds for a class of \emph{random} partial differential equations and \emph{stochastic} partial differential equations is shown. Unlike the invariant manifold theory for stochastic \emph{ordinary} differential equations, random norms are not used. The result is then applied to a nonlinear stochastic partial differential equation with linear multiplicative noise.

math.DS↗

arXiv2008

Predictability in Nonlinear Dynamical Systems with Model Uncertainty

Nonlinear systems with model uncertainty are often described by stochastic differential equations. Some techniques from random dynamical systems are discussed. They are relevant to better understanding of solution processes of stochastic differential equations and thus may shed lights on predictability in nonlinear systems with model uncertainty.

math.DS↗

arXiv2008

Stochastic Modeling of Unresolved Scales in Complex Systems

Model uncertainties and simulation uncertainties occur in mathematical modeling of multiscale complex systems, since some mechanisms or scales are not represented (i.e., "unresolved") due to lack in our understanding of these mechanisms or limitations in computational power. The impact of these unresolved scales on the resolved scales needs to be parameterized or taken into account. A stochastic scheme is devised to take the effects of unresolved scales into account, in the context of solving nonlinear partial differential equations. An example is presented to demonstrate this strategy.

math.DS↗

arXiv2008

Large deviations for the Boussinesq Equations under Random Influences

A Boussinesq model for the Benard convection under random influences is considered as a system of stochastic partial differential equations. This is a coupled system of stochastic Navier-Stokes equations and the transport equation for temperature. Large deviations are proved, using a weak convergence approach based on a variational representation of functionals of infinite dimensional Brownian motion.

math.PR↗