Reductions and deviations for stochastic partial differential equations under fast dynamical boundary conditions

In order to understand the impact of random influences at physical boundary on the evolution of multiscale systems, a stochastic partial differential equation model under a fast random dynamical boundary condition is investigated. The noises in the model and in the boundary condition are both additive. An effective equation is derived and justified by reducing the random \emph{dynamical} boundary condition to a simpler one. The effective system is still a stochastic partial differential equation. Furthermore, the quantitative comparison between the solution of the original stochastic system and the effective solution is provided by establishing normal deviations and large deviations principles. Namely, the normal deviations are asymptotically characterized, while the rate and speed of the large deviations are estimated.Comment: This is a revised version with 29 pages. To appear in Stochastic Analysis and Applications, 200

Geometric Methods for Stochastic Dynamical Systems

Noisy fluctuations are ubiquitous in complex systems. They play a crucial or delicate role in the dynamical evolution of gene regulation, signal transduction, biochemical reactions, among other systems. Therefore, it is essential to consider the effects of noise on dynamical systems. It has been a challenging topic to have better understanding of the impact of the noise on the dynamical behaviors of complex systems.Comment: 8 page

Mean exit time for stochastic dynamical systems driven by tempered stable L\'evy fluctuations

We use the mean exit time to quantify macroscopic dynamical behaviors of stochastic dynamical systems driven by tempered L\'evy fluctuations, which are solutions of nonlocal elliptic equations. Firstly, we construct a new numerical scheme to compute and solve the mean exit time associated with the one dimensional stochastic system. Secondly, we extend the analytical and numerical results to two dimensional case: horizontal-vertical and isotropic case. Finally, we verify the effectiveness of the presented schemes with numerical experiments in several examples

Dynamics of the Thermohaline Circulation Under Uncertainty

The ocean thermohaline circulation under uncertainty is investigated by a random dynamical systems approach. It is shown that the asymptotic dynamics of the thermohaline circulation is described by a random attractor and by a system with finite degrees of freedom.Comment: 15 page

Compactly Generated Shape Index Theory and its Application to a Retarded Nonautonomous Parabolic Equation

We establish the compactly generated shape (H-shape) index theory for local semiflows on complete metric spaces via more general shape index pairs, and define the H-shape cohomology index to develop the Morse equations. The main advantages are that the quotient space $N/E$ is not necessarily metrizable for the shape index pair $(N,E)$ and N\sm E need not to be a neighborhood of the compact invariant set. Moreover, in this new theory, the phase space is not required to be separable. We apply H-shape index theory to an abstract retarded nonautonomous parabolic equation to obtain the existence of bounded full solutions

Ergodic Dynamics of the Stochastic Swift-Hohenberg System

The Swift-Hohenberg fluid convection system with both local and nonlocal nonlinearities under the influence of white noise is studied. The objective is to understand the difference in the dynamical behavior in both local and nonlocal cases. It is proved that when sufficiently many of its Fourier modes are forced, the system has a unique invariant measure, or equivalently, the dynamics is ergodic. Moreover, it is found that the number of modes to be stochastically excited for ensuring the ergodicity in the local Swift-Hohenberg system depends {\em only} on the Rayleigh number (i.e., it does not even depend on the random term itself), while this number for the nonlocal Swift-Hohenberg system relies additionally on the bound of the kernel in the nonlocal interaction (integral) term, and on the random term itselfComment: Version: Oct 9, 2003; accepted August 18, 200

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.Comment: 30 page

Most Probable Evolution Trajectories in a Genetic Regulatory System Excited by Stable L\'evy Noise

We study the most probable trajectories of the concentration evolution for the transcription factor activator in a genetic regulation system, with non-Gaussian stable L\'evy noise in the synthesis reaction rate taking into account. We calculate the most probable trajectory by spatially maximizing the probability density of the system path, i.e., the solution of the associated nonlocal Fokker-Planck equation. We especially examine those most probable trajectories from low concentration state to high concentration state (i.e., the likely transcription regime) for certain parameters, in order to gain insights into the transcription processes and the tipping time for the transcription likely to occur. This enables us: (i) to visualize the progress of concentration evolution (i.e., observe whether the system enters the transcription regime within a given time period); (ii) to predict or avoid certain transcriptions via selecting specific noise parameters in particular regions in the parameter space. Moreover, we have found some peculiar or counter-intuitive phenomena in this gene model system, including (a) a smaller noise intensity may trigger the transcription process, while a larger noise intensity can not, under the same asymmetric L\'evy noise. This phenomenon does not occur in the case of symmetric L\'evy noise; (b) the symmetric L\'evy motion always induces transition to high concentration, but certain asymmetric L\'evy motions do not trigger the switch to transcription. These findings provide insights for further experimental research, in order to achieve or to avoid specific gene transcriptions, with possible relevance for medical advances

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 L\'evy noise is present, the conventional Kalman filter may fail to be effective due to the fact that the non-Gaussian L\'evy noise may have infinite variance. A modified Kalman filter for linear systems with non-Gaussian L\'evy noise is devised. It works effectively with reasonable computational cost. Simulation results are presented to illustrate this non-Gaussian filtering method

Joint Background Reconstruction and Foreground Segmentation via A Two-stage Convolutional Neural Network

Foreground segmentation in video sequences is a classic topic in computer vision. Due to the lack of semantic and prior knowledge, it is difficult for existing methods to deal with sophisticated scenes well. Therefore, in this paper, we propose an end-to-end two-stage deep convolutional neural network (CNN) framework for foreground segmentation in video sequences. In the first stage, a convolutional encoder-decoder sub-network is employed to reconstruct the background images and encode rich prior knowledge of background scenes. In the second stage, the reconstructed background and current frame are input into a multi-channel fully-convolutional sub-network (MCFCN) for accurate foreground segmentation. In the two-stage CNN, the reconstruction loss and segmentation loss are jointly optimized. The background images and foreground objects are output simultaneously in an end-to-end way. Moreover, by incorporating the prior semantic knowledge of foreground and background in the pre-training process, our method could restrain the background noise and keep the integrity of foreground objects at the same time. Experiments on CDNet 2014 show that our method outperforms the state-of-the-art by 4.9%.Comment: ICME 201