arXiv subjects

Jinqiao Duan

Publications and source records attributed to Jinqiao Duan.

At least 181 records · Page 10Linked to original sources

arXiv2015

Dynamical Inference for Transitions in Stochastic Systems with $α-$stable Lévy Noise

A goal of data assimilation is to infer stochastic dynamical behaviors with available observations. We consider transition phenomena between metastable states for a stochastic system with (non-Gaussian) $α-$stable Lévy noise. With either discrete time or continuous time observations, we infer such transitions by computing the corresponding nonlocal Zakai equation (and its discrete time counterpart) and examining the most probable orbits for the state system. Examples are presented to demonstrate this approach.

math.DS↗

arXiv2015

A regularity result for the nonlocal Fokker-Planck equation with Ornstein-Uhlenbeck drift

Despite there are numerous theoretical studies of stochastic differential equations with a symmetric $α$-stable Lévy noise, very few regularity results exist in the case of $0<α\leq1$. In this paper, we study the fractional Fokker-Planck equation with Ornstein-Uhlenbeck drift, and prove that there exists a unique solution, which is $C^\infty$ in space for $t>0$ when $α\in (0, 2]$.

math.DS↗

arXiv2014

Martingale and Weak Solutions for a Stochastic Nonlocal Burgers Equation on Bounded Intervals

This work is about the existence of martingale solutions and weak solutions for a stochastic nonlocal Burgers equation on bounded intervals. The existence of a martingale solution is shown by using a Galerkin approximation, Prokhorov's theorem and Skorokhod's embedding theorem. The same Galerkin approximation also leads to the existence of weak solution for the corresponding deterministic nonlocal Burgers equation on a bounded domain.

math.PR↗

arXiv2014

Impacts of Noise on a Class of Partial Differential Equations

This paper is concerned with effects of noise on the solutions of partial differential equations. We first provide a sufficient condition to ensure the existence of a unique positive solution for a class of stochastic parabolic equations. Then, we prove that noise could induce singularities (finite time blow up of solutions). Finally, we show that a stochastic Allen-Cahn equation does not have finite time singularities and the unique solution exists globally.

math.AP↗

arXiv2014

Approximation of the inertial manifold for a nonlocal dynamical system

We consider inertial manifolds and their approximation for a class of partial differential equations with a nonlocal Laplacian operator $-(-Δ)^{\fracα{2}}$, with $0<α<2$. The nonlocal or fractional Laplacian operator represents an anomalous diffusion effect. We first establish the existence of an inertial manifold and highlight the influence of the parameter $α$. Then we approximate the inertial manifold when a small normal diffusion $\varepsilon Δ$ (with $\varepsilon \in (0, 1)$) enters the system, and obtain the estimate for the Hausdorff semi-distance between the inertial manifolds with and without normal diffusion.

math.AP↗

arXiv2014

Asymptotic methods for stochastic dynamical systems with small non-Gaussian Lévy noise

The goal of the paper is to analytically examine escape probabilities for dynamical systems driven by symmetric $α$-stable Lévy motions. Since escape probabilities are solutions of a type of integro-differential equations (i.e., differential equations with nonlocal interactions), asymptotic methods are offered to solve these equations to obtain escape probabilities when noises are sufficiently small. Three examples are presented to illustrate the asymptotic methods, and asymptotic escape probability is compared with numerical simulations.

math.PR↗

arXiv2014

Stationary Measures for Stochastic Differential Equations with Jumps

In the paper, stationary measures of stochastic differential equations with jumps are considered. Under some general conditions, existence of stationary measures is proved through Markov measures and Lyapunov functions. Moreover, for two special cases, stationary measures are given by solutions of Fokker-Planck equations and long time limits for the distributions of system states.

math.PR↗

arXiv2013

Global solutions for a nonlocal Ginzberg-Landau equation and a nonlocal Fokker-Plank equation

This work is devoted to the study of a nonlocal Ginzberg-Landau equation by the semigroup method and a nonlocal Fokker-Plank equation by the viscosity vanishing method. For the nonlocal Ginzberg-Landau equation, there exists a unique global solution in the set $C^0(\mathbb{R}^+,\,H_0^{\fracα{2}}(D))\cap L_{loc}(\mathbb{R}^+,\,H_0^α(D))$, for $α\in (0,\,2)$. For the nonlocal Fokker-Plank equation, the regularity of the solution is weaker than that of the nonlocal Ginzberg-Landau equation due to the drift term.

math.AP↗

arXiv2013

Slow foliation of a slow-fast stochastic evolutionary system

This work is concerned with the dynamics of a slow-fast stochastic evolutionary system quantified with a scale parameter. An invariant foliation decomposes the state space into geometric regions of different dynamical regimes, and thus helps understand dynamics. A slow invariant foliation is established for this system. It is shown that the slow foliation converges to a critical foliation (i.e., the scale parameter is zero) in probability distribution, as the scale parameter tends to zero. The approximation of slow foliation is also constructed with error estimate in distribution. Furthermore, the geometric structure of the slow foliation is investigated: every fiber of the slow foliation parallels each other, with the slow manifold as a special fiber. In fact, when an arbitrarily chosen point of a fiber falls in the slow manifold, the fiber must be the slow manifold itself.

math.AP↗

arXiv2013

Fokker-Planck Equations for Stochastic Dynamical Systems with Symmetric Lévy Motions

The Fokker-Planck equations for stochastic dynamical systems, with non-Gaussian $α-$stable symmetric Lévy motions, have a nonlocal or fractional Laplacian term. This nonlocality is the manifestation of the effect of non-Gaussian fluctuations. Taking advantage of the Toeplitz matrix structure of the time-space discretization, a fast and accurate numerical algorithm is proposed to simulate the nonlocal Fokker-Planck equations, under either absorbing or natural conditions. The scheme is shown to satisfy a discrete maximum principle and to be convergent. It is validated against a known exact solution and the numerical solutions obtained by using other methods. The numerical results for two prototypical stochastic systems, the Ornstein-Uhlenbeck system and the double-well system are shown.

math.NA↗

arXiv2013

Center manifolds for infinite dimensional random dynamical systems

Stochastic center manifolds theory are crucial in modelling the dynamical behavior of complex systems under stochastic influences. A multiplicative ergodic theorem on Hilbert space is proved to be satisfied to the exponential trichotomy condition. Then the existence of stochastic center manifolds for infinite dimensional random dynamical systems is shown under the assumption of exponential trichotomy. The theory provides a support for the discretisations of nonlinear stochastic partial differential equations with space-time white noise.

math.DS↗

arXiv2013

Slow Manifolds for Multi-Time-Scale Stochastic Evolutionary Systems

This article deals with invariant manifolds for infinite dimensional random dynamical systems with different time scales. Such a random system is generated by a coupled system of fast-slow stochastic evolutionary equations. Under suitable conditions, it is proved that an exponentially tracking random invariant manifold exists, eliminating the fast motion for this coupled system. It is further shown that if the scaling parameter tends to zero, the invariant manifold tends to a slow manifold which captures long time dynamics. As examples the results are applied to a few systems of coupled parabolic-hyperbolic partial differential equations, coupled parabolic partial differential-ordinary differential equations, and coupled hyperbolic-hyperbolic partial differential equations.

math.PR↗

arXiv2013

Quantifying model uncertainty in non-Gaussian dynamical systems with observations on mean exit time or escape probability

Complex systems are sometimes subject to non Gaussian alpha stable Levy fluctuations. A new method is devised to estimate this uncertain parameter and other system parameters, using observations on either mean exit time or escape probability for the system evolution. It is based on solving an inverse problem for a deterministic, non-local partial differential equation via numerical optimization. The existing methods for estimating parameters require observations on system state sample paths for long time periods or probability densities at large spatial ranges. The method proposed here, instead, requires observations on mean exit time or escape probability only for an arbitrarily small spatial domain. This new method is beneficial to systems for which mean exit time or escape probability is feasible to observe.

math.DS↗

arXiv2013

A parameter estimation method based on random slow manifolds

A parameter estimation method is devised for a slow-fast stochastic dynamical system, where often only the slow component is observable. By using the observations only on the slow component, the system parameters are estimated by working on the slow system on the random slow manifold. This offers a benefit of dimension reduction in quantifying parameters in stochastic dynamical systems. An example is presented to illustrate this method, and verify that the parameter estimator based on the lower dimensional, reduced slow system is a good approximation of the parameter estimator for original slow-fast stochastic dynamical system.

math.DS↗

arXiv2013

Approximation of Random Slow Manifolds and Settling of Inertial Particles under Uncertainty

A method is provided for approximating random slow manifolds of a class of slow-fast stochastic dynamical systems. Thus approximate, low dimensional, reduced slow systems are obtained analytically in the case of sufficiently large time scale separation. To illustrate this dimension reduction procedure, the impact of random environmental fluctuations on the settling motion of inertial particles in a cellular flow field is examined. It is found that noise delays settling for some particles but enhances settling for others. A deterministic stable manifold is an agent to facilitate this phenomenon. Overall, noise appears to delay the settling in an averaged sense.

math.DS↗

arXiv2013

State estimation under non-Gaussian Levy noise: A modified Kalman filtering method

The Kalman filter is extensively used for state estimation for linear systems under Gaussian noise. When non-Gaussian Lévy noise is present, the conventional Kalman filter may fail to be effective due to the fact that the non-Gaussian Lévy noise may have infinite variance. A modified Kalman filter for linear systems with non-Gaussian Lévy noise is devised. It works effectively with reasonable computational cost. Simulation results are presented to illustrate this non-Gaussian filtering method.

math.DS↗

arXiv2013

Stochastic Averaging Principle for Dynamical Systems with Fractional Brownian Motion

Stochastic averaging for a class of stochastic differential equations (SDEs) with fractional Brownian motion, of the Hurst parameter H in the interval (1/2, 1), is investigated. An averaged SDE for the original SDE is proposed, and their solutions are quantitatively compared. It is shown that the solution of the averaged SDE converges to that of the original SDE in the sense of mean square and also in probability. It is further demonstrated that a similar averaging principle holds for SDEs under stochastic integral of pathwise backward and forward types. Two examples are presented and numerical simulations are carried out to illustrate the averaging principle.

math.DS↗

arXiv2012

Synchronization of an evolving complex hyper-network

In this paper, the synchronization in a hyper-network of coupled dynamical systems is investigated for the first time. An evolving hyper-network model is proposed for better describing some complex systems. A concept of joint degree is introduced, and the evolving mechanism of hyper-network is given with respect to the joint degree. The hyper-degree distribution of the proposed evolving hyper-network is derived based on a rate equation method and obeys a power law distribution. Furthermore, the synchronization in a hyper-network of coupled dynamical systems is investigated for the first time. By calculating the joint degree matrix, several simple yet useful synchronization criteria are obtained and illustrated by several numerical examples.

nlin.AO↗