On Uniqueness of Maximal Coupling for Diffusion Processes with a Reflection

A maximal coupling of two diffusion processes makes two diffusion particles meet as early as possible. We study the uniqueness of maximal couplings under a sort of ‘reflection structure’ which ensures the existence of such couplings. In this framework, the uniqueness in the class of Markovian couplings holds for the Brownian motion on a Riemannian manifold whereas it fails in more singular cases. We also prove that a Kendall-Cranston coupling is maximal under the reflection structure.     

关于多维扩散过程耦合的保序性

本文给出了多维扩散过程耦合保序的若干充分条件和必要条件，特别地，当耦合过程存在唯一时，得到了耦合保序的充要条件。     

非时齐马氏过程的耦合基本定理

本文将耦合方法应用于非时齐马氏过程,推广了时齐情形的耦合基本定理,为后续研究非时齐马氏过程的耦合提供了理论基础.     

紧流形上非退化扩散过程的成功耦合

本文构造了紧连通流形上非退化扩散过程的一些耦合，并使用随机分析的方法证明这些耦合是成功的，进一步地，耦合时间一阶矩有限。最后，我们使用耦合方法研究了扩散过程依全变差范数的收敛速度。     

Monotonicity of Time-Dependent Transportation Costs and Coupling by Reflection

Based on a study of the coupling by reflection of diffusion processes, a new monotonicity in time of a time-dependent transportation cost between heat distribution is shown under Bakry-Émery’s curvature-dimension condition on a Riemannian manifold. The cost function comes from the total variation between heat distributions on spaceforms. As a corollary, we obtain a comparison theorem for the total variation between heat distributions. In addition, we show that our monotonicity is stable under the Gromov-Hausdorff convergence of the underlying space under a uniform curvature-dimension and diameter bound.     

Mirror and synchronous couplings of geometric Brownian motions

The paper studies the question of whether the classical mirror and synchronous couplings of two Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We establish a characterisation of the optimality of the two couplings over any finite time horizon and show that, unlike in the case of Brownian motion, the optimality fails in general even if the geometric Brownian motions are martingales. On the other hand, we prove that in the cases of the ergodic average and the infinite time horizon criteria, the mirror coupling and the synchronous coupling are always optimal for general (possibly non-martingale) geometric Brownian motions. We show that the two couplings are efficient if and only if they are optimal over a finite time horizon and give a conjectural answer for the efficient couplings when they are suboptimal.     

Finite-time stochastic synchronization of complex networks

In this paper, we study the finite-time stochastic synchronization problem for complex networks with stochastic noise perturbations. By using finite-time stability theorem, inequality techniques, the properties of Weiner process and adding suitable controllers, sufficient conditions are obtained to ensure finite-time stochastic synchronization for the complex networks. The effects of control parameters on synchronization speed and time are also analyzed. The results of this paper are applicable to both directed and undirected weighted networks while do not need to know any information about eigenvalues of coupling matrix. Since finite-time synchronization means the optimality in convergence time and has better robustness and disturbance rejection properties, the results of this paper are important. A numerical example shows the effectiveness of our new results.     

Existence of the optimal measurable coupling and ergodicity for Markov processes

The existence theorem of the optimal measurable coupling of two probability kernels on a complete separable metric measurable space is proved. Then by this theorem, a general ergodicity theorem for Markov processes is obtained. And as an immediate application to particle systems the uniqueness theorem of the stationary distribution is supplemented, i.e. the uniqueness theorem also implies the existence of the stationary distribution.     

Ergodicity for neutral type SDEs with infinite length of memory

In this paper, the weak Harris theorem developed in [18] is illustrated by using a straightforward Wasserstein coupling, which implies the exponential ergodicity of the functional solutions to a range of neutral type SDEs with infinite length of memory. A concrete example is presented to illustrate the main result.     

Stabilization problem of stochastic time-varying coupled systems with time delay and feedback control

The stabilization of stochastic coupled systems with time delay and time-varying coupling structure (SCSTT) via feedback control is investigated. We generalize systems with constant coupling structure to the time-varying coupling structure. Combining the graph theory with the Lyapunov method, a systematic method is provided to construct a Lyapunov function for SCSTT, and a Lyapunov-type theorem and a coefficient-type criterion are obtained to guarantee the stabilization in the sense of pth moment exponential stability. Furthermore, theoretical results are applied to analyze the stabilization of stochastic-coupled oscillators with time delay and time-varying coupling structure in order to illustrate the practicability of the results. Finally, two numerical examples are given to illustrate the effectiveness and feasibility of theoretical results.     

Globally Convergent,Iterative Path-Following for Algebraic Equations

Homotopy methods are of great importance for the solution of systems of equations. It is a major problemto ensure well-defined iterations along the homotopy path. Many investigations have considered the complexityof path-following methods depending on the unknown distance of some given path to the variety of ill-posed problems.It is shown here that there exists a construction method for safe paths for a single algebraic equation. A safepath may be effectively determined with bounded effort. Special perturbation estimates for the zeros together withconvergence conditions for Newton’s method in simultaneous mode allow our method to proceed on the safe path.This yields the first globally convergent, never-failing, uniformly iterative path-following algorithm. The maximumnumber of homotopy steps in our algorithm reaches a theoretical bound forecast by Shub and Smale i.e., the numberof steps is at most quadratic in the condition number. A constructive proof of the fundamental theorem of algebrameeting demands by Gauß, Kronecker and Weierstraß is a consequence of our algorithm.     

Successful couplings for a class of stochastic differential equations driven by Lvy processes

By constructing proper coupling operators for the integro-differential type Markov generator,we establish the existence of a successful coupling for a class of stochastic differential equations driven by L’evy processes.Our result implies a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying Markov semigroups,and it is sharp for Ornstein-Uhlenbeck processes driven by α-stable L’evy processes.     

一类非线性演化方程族的可积耦合

Starting from a Tu Guizhang‘s isospectral‘problem, a Lax pair is obtained by means of Tu scheme ( we call it Tu Lax pair ). By applying a gauge transformation between matrices, the Tu Lax pair is changed to its equivalent Lax pair with the traces of spectral matrices being zero, whose compatibility gives rise to a type of Tu hierarchy of equations. By making use of a high order loop algebra constructed by us, an integrable coupling system of the Tu hierarchy of equations are presented. Especially, as reduction cases, the integrable couplings of the celebrated AKNS hierarchy, TD hierarchy and Levi hierarchy are given at the same time.     

A non-vanishing property for the signature of a path

We prove that a continuous path with finite length in a real Banach space cannot have infinitely many zero components in its signature unless it is tree-like. In particular, this allows us to strengthen a limit theorem for signature recently proved by Chang, Lyons, and Ni. What lies at the heart of our proof is a complexification idea together with deep results from holomorphic polynomial approximations in the theory of several complex variables.     

关于非负下半连续函数最优可测耦合的存在性及其应用

本文证明了两个转移概率关于非负下半连续函数最优可测耦合的存在性定理．作为对这一结果的应用,推广了Ｓｔｒａｓｓｅｎ定理,进而证明了跳过程的随机可比性等价于保序耦合的存在性．     

Synchronization analysis of complex‐Variable chaotic systems with discontinuous unidirectional coupling

This article is concerned with the problem of synchronization between two uncertain complex‐variable chaotic systems with parameters perturbation and discontinuous unidirectional coupling. Based on the stability theory and comparison theorem of differential equations, some sufficient conditions for the complete synchronization and generalized synchronization are obtained. The theoretical results show that the two uncertain complex‐variable chaotic systems with discontinuous unidirectional coupling can achieve synchronization if the time‐average coupling strength is large enough. Finally, numerical examples are examined to illustrate the feasibility and effectiveness of the analytical results. © 2015 Wiley Periodicals, Inc. Complexity 21: 343–355, 2016     

Multi-agent system model with mixed coupling topologies for pattern formation and formation splitting

In this article, a new type of multi-agent system model with mixed coupling topologies is proposed for realizing pattern formations with specific geometric shapes and formation splitting. The interactions among individual agents are assumed to be universally repulsive and selectively attractive. By designing the form of attractive coupling matrix, one can obtain a variety of formations with specific shapes in the system through self-assembly of agents. Both symmetric coupling case and asymmetric coupling case are considered. Analysis and simulation results show symmetric ones result in convergent dynamics to steady-state formations, whereas, for asymmetric case, the system exhibits complex dynamic behaviours, including collective rotation and chaotic motion. By breaking the graph defined by attractive couplings into disjoint subgraphs, one can make the formation of agents to split into small sizes. The results are relevant for the design of coordination and cooperative control for multi-agent systems.     

A primal-dual homotopy algorithm for $$\ell _{1}$$-minimization with $$\ell _{\infty }$$ℓ∞-constraints

In this paper we propose a primal-dual homotopy method for \(\ell _1\)-minimization problems with infinity norm constraints in the context of sparse reconstruction. The natural homotopy parameter is the value of the bound for the constraints and we show that there exists a piecewise linear solution path with finitely many break points for the primal problem and a respective piecewise constant path for the dual problem. We show that by solving a small linear program, one can jump to the next primal break point and then, solving another small linear program, a new optimal dual solution is calculated which enables the next such jump in the subsequent iteration. Using a theorem of the alternative, we show that the method never gets stuck and indeed calculates the whole path in a finite number of steps. Numerical experiments demonstrate the effectiveness of our algorithm. In many cases, our method significantly outperforms commercial LP solvers; this is possible since our approach employs a sequence of considerably simpler auxiliary linear programs that can be solved efficiently with specialized active-set strategies.     

On The Existence Of A Measurable Optimal Coupling Of Probability Kernels

In this paper, an existence theorem for a measurable optimal coupling of probability kernels on a σ-compact complete separable metric measurable space is obtained     

Exponential Convergence Rates of Markov Chains under a Weaken Minorization Condition

In this paper, we obtain the quantitative bound of the exponential convergence rates of Markov chains under a weaken minorization condition, using the coupling method and the analytic approach. And also, we obtain the convergence rates for continuous time Markov processes.