召集人：吕克宁（四川大学）、段金桥（大湾区大学）、刘培东（北京大学）

时间：2024.08.04—2024.08.10

随机动力系统研讨会

会议日程

2024年8月5日（星期一）  天元数学国际交流中心 

2024年8月6日（星期二）  天元数学国际交流中心   

2024年8月7日（星期三）  天元数学国际交流中心     

2024年8月8日（星期四）  天元数学国际交流中心

2024年8月9日（星期五）  天元数学国际交流中心     

学术报告信息

Asymptotic behaviors for fractional SPDEs on unbounded domains

陈章（山东大学）

摘要：In this talk, long time dynamics and limiting behavior for stochastic reaction-diffusion equations with fractional Laplacian on unbounded domains will be mainly introduced, which will include invariant measures, periodic measures and large deviations.

Synchronization of Stochastic Differential Equations

高洪俊（东南大学）

摘要: In this talk, I will talk about pathwise synchronization of global coupled system with linear multiplicative rough noise and synchronization of stochastic differential equations driven by nonlinear multiplicative noise in mean square.

Polynomial mixing for white-forced Kuramoto-Sivashinsky equation on the whole line

高鹏（东北师范大学）

摘要：In this talk, I want to show some new developments on ergodicity of white-forced Kuramoto-Sivashinsky equation on the whole line.

Numerical Ergodicity of Stochastic Burgers-Huxley Equation and 2D Navier-Stokes Equations

黄建华 (国防科技大学)

摘要：In this talk, we present some results about the numerical ergodicity of stochastic Burgers-Huxley equation and 2D Navier-Stokes equations driven by Gaussian noise. The ergodicity and convergence order are established using spatial Galerkin semi-discretization firstly. The convergence order of the numerical invariant measures under the exponential Euler spatiotemporal complete discretization, which yields a numerical ergodic measure, is demonstrated to be 1/2. Finally, numerical simulations are offered to validate the theoretical findings.

Probability Flow Approach to the Onsager--Machlup Functional for Jump-Diffusion Processes

         黄远飞（香港城市大学）

摘要：The classical Onsager--Machlup method is fundamental in the analysis of Gaussian stochastic dynamical systems and is widely used across various fields. However, applying it directly to jump-diffusion processes presents significant technical challenges due to the complexities introduced by jump noise. From a path integral perspective, the transition density becomes highly intricate, making it difficult to provide a clear and convergent functional representation of smooth paths. Moreover, from a probabilistic standpoint, the traditional approach of changing measures using the Girsanov theorem requires aggregating all trajectories within a predefined tube, demanding uniform properties across these trajectories—properties that jump counts in jump-diffusion processes inherently lack. To address these challenges, we propose a novel methodology that leverages the probabilistic flow equivalence between jump-diffusions and pure diffusions. We have derived a closed-form expression for the Onsager--Machlup functional for jump-diffusion processes with finite jump activity by approximating both path integrals and probabilistic approaches. Our analysis shows that the impact of jump noise on the Onsager--Machlup functional is primarily determined by the behavior of the L\'{e}vy intensity at the initial point. Additionally, we investigate scenarios involving jump-diffusion processes with infinite jump activity. Although establishing a well-defined Onsager--Machlup functional for these cases remains challenging, we provide a discrete version as a practical alternative.

On the Exponential Ergodicity and Large and Moderate Deviations of Stochastic Relaxation Damping Hamiltonian Systems

蒋继发（上海师范大学）

摘要：This talk will first give a new criterion on the existence of  periodic solution in distribution for time-periodic stochastic Hamiltonian systems with relaxation damping and possibly bounded force by the potential and unbounded random force. The criteria on the exponential ergodicity 、large deviation principles of Donsker and Varadhan (LDP)、 moderate deviation principles and Freidlin-Wentzell type asymptotic measure concentration theorem are presented. The key techniques are to construct better Lyapunov functions based on Wu liming’s theory on stochastic Hamiltonian systems.

Stability of smooth solitary wave of the Degasperis-Procesi and the modified Camassa-Holm equation

李骥 （华中科技大学）

 摘要：We first review some aspects of soliton for the KdV equation. Then we introduce a class of quasilinear shallow water equations: Camassa-Holm(CH), Degasperis-Procesi (DP), and the modified Camassa-Holm(mCH). We explain how their solitons could be proved stable by Lyapunov functional methods or by Variational method. We also give a first result on global wellposedness of the mCH equation.

Prevalent and typical properties of the local entropy function

历智明 （西北大学）

摘要：In this talk, we investigate the local entropy function of topological dynamical systems.

We show that for a prevalent measure within the set of all Borel probability measures, the infimum of upper and lower measure-theoretic entropy is attainable and equals zero. While the local entropy function of a continuous flow for a typical measure is incredibly complex and irregular, and even after employing widely accepted and significantly effective smoothing techniques, including higher-order Riesz-Hardy logarithmic averages and Cesro averages. That is, the lower average local entropy function is zero, while the upper average local entropy function tends to infinity.

Multifractal analysis for a class of skew product transformations

刘学（东南大学）

摘要：In this talk, we give a multifractal analysis on the Birkhoff average for a class of skew product transformations, which is driven by a uniquely ergodic homeomorphism and satisfies Anosov and topological mixing on fibers property. The conclusion is twofold: a variational principle between the fiber Bowen’s topological entropy on conditional level sets of Birkhoff average and fiber measure-theoretical entropy; If the irregular set is nonempty on some fibers, then on almost every fiber with respect to the unique ergodic measure, it is nonempty and carries full fiber Bowen’s topological entropy. Examples of systems under consideration are provided. This is a joint work with Nian Liu.

The locally homeomorphic property and the multiplicative ergodic theorem for McKean-Vlasov SDEs

柳振鑫（大连理工大学）

摘要：In this talk, we will explore two aspects that distinguish McKean-Vlasov SDEs significantly from classical SDEs. In the first part, we establish the locally diffeomorphic property of the solution to McKean-Vlasov SDEs defined in the Euclidean space. We observe that although the coefficients are global Lipschitz, the solution in general does not satisfy the globally homeomorphic property at any time except the initial time. In the second part, we introduce the concept of Lyapunov exponents for McKean-Vlasov SDEs. We observe that even when the coefficients are regular enough and the first-order derivatives are bounded, the limit in the definition of Lyapunov exponents may not exist. Furthermore, we establish the mean-field version of the multiplicative ergodic theorem. This talk is based on the collaboration with Xianjin Cheng and Lixin Zhang.

Almost Sure Averaging for Fast-slow Stochastic Differential Equations via Controlled Rough Path

裴斌 （西北工业大学）

摘要：This talk focuses on the averaging method to a coupled system consisting of two stochastic differential equations which has a slow component driven by fractional Brownian motion (FBM) with less regularity $1/3< H \leq 1/2$ and a fast dynamics under additive FBM with Hurst-index $1/3< \hat H \leq 1/2$. We prove that the solution of the slow component converges almost surely to the solution of the corresponding averaged equation using the approach of time discretization and controlled rough path. To do this, we employ the random dynamical system (RDS) to obtain a stationary solution by an exponentially attracting random fixed point of the RDS generated by the non-Markovian fast component.

Spectral rigidity and joint integrability for Anosov systems on tori

史逸（四川大学）

摘要：In this talk, we address on the strong rigidity properties from joint integrability in the setting of Anosov diffeomorphisms on tori. More specifically, for an irreducible Anosov diffeomorphism with splitted stable bundle, the joint integrability of the strong stable and full unstable subbundles implies existence of fine dominated splitting along the weak stable subbundle as well as Lyapunov exponents rigidity. This builds an equivalence bridge between the geometric rigidity (joint integrability) and dynamical spectral rigidity (Lyapunov exponents rigidity) for Anosov diffeomorphisms on tori. Moreover, we show that if two non-invertible Anosov maps on 2-torus are topological conjugate, then they also admit spectral rigidity along stable bundles, i.e. they have the same Lyapunov exponents on corresponding periodic points. In particular, the conjugacy is smooth along stable foliation.

负曲率流形上的（随机）动力学

束琳 （北京大学）

摘要：如何运用动力学性质刻画空间的几何性质是动力系统的基础研究课题之一。在这个报告中，我们将聚焦负曲率流形上的（随机）动力系统，简介“熵”、 “Lyapunov 指数”等动力系统量与空间几何性质之间的联系和刻画。

Emergent dynamics in mean filed limit

王伟（南京大学）

摘要：In this talk I will present a new idea to study the dynamics of the McKean Vlasov equations and shows why different dynamics emerges in mean filed limit.

Two-Time-Scale Stochastic Functional Differential Equations: Inclusion of Infinite Delay and Coupled Segment Processes

吴付科（华中科技大学）

摘要：This paper focuses on two-time-scale stochastic functional differential equations (SFDEs). It features in inclusion of infinite delay and coupling of  slow and fast components. The coupling is through the segment processes of the slow and fast processes. The main difficulties include infinite delay and the coupling of  segment processes in the fast and slow motions. Concentrating on weak convergence, tightness of the segment process is established on a space of continuous functions. In addition, the Holder continuity and boundedness for the segment process of the slow component, uniform boundedness for the segment process of a fixed-$x$ SFDE, exponential ergodicity, and continuous dependence on parameters are obtained to carry out the desired asymptotic analysis, and also as  byproducts, which are interesting in their own right. Then using the martingale problem formulation, an average principle is  established by a direct averaging, which involves complex computations and subtle estimates. Finally, as two classes of special SFDEs, stochastic integro-differential equations and stochastic delay differential equations with two-time scales are investigated.

A stochastic nutrient-phytoplankton-zooplankton-model and its asymptotic behaviors

袁荣 （北京师范大学）

摘要：In this talk, we consider the stochastic nutrient-phytoplankton-zooplankton model with nutrient cycle. In order to take stochastic fluctuations into account, we add the stochastic increments to the variations of biomass of nutrition, phytoplankton and zooplankton during time interval Δt, thus we obtain the corresponding stochastic model. Subsequently, we explore the existence, uniqueness and stochastically ultimate boundness of global positive solution. By constructing suitable Lyapunov function, we also obtain V -geometric ergodicity of this model. In addition, the sufficient conditions of exponential extinction and persistence in the mean of plankton are established. At last, we present some numerical simulations to validate theoretical results and analyse the impacts of some important parameters.

Most Probable Dynamics of the Single-Species with Allee Effect under Jump-Diffusion Noise

袁胜兰 (大湾区大学)

摘要：We explore the most probable phase portrait (MPPP) of a stochastic single-species model incorporating the Allee effect by utilizing the nonlocal Fokker-Planck equation (FPE). This stochastic model incorporates both non-Gaussian and Gaussian noise sources. It has three fixed points in the deterministic case. One is the unstable state, which lies between the two stable equilibria. Our primary focus is on elucidating the transition pathways from extinction to the upper stable state in this single-species model, particularly under the influence of jump-diffusion noise. This helps us to study the biological behavior of species. The identification of the most probable path relies on solving the nonlocal FPE tailored to the population dynamics of the single-species model. This enables us to pinpoint the corresponding maximum possible stable equilibrium state. Additionally, we derive the Onsager-Machlup function for the stochastic model and employ it to determine the corresponding most probable paths. Numerical simulations manifest three key insights: (i) when non-Gaussian noise is present in the system, the peak of the stationary density function aligns with the most probable stable equilibrium state; (ii) if the initial value rises from extinction to the upper stable state, then the most probable trajectory converges towards the maximally probable equilibrium state, situated approximately between 9 and 10; and (iii) the most probable paths exhibit a rapid ascent towards the stable state, then maintain a sustained near-constant level, gradually approaching the upper stable equilibrium as time goes on These numerical findings pave the way for further experimental investigations aiming to deepen our comprehension of dynamical systems within the context of biological modeling.

分数Brown运动驱动的随机动力系统的复杂动力学

曾才斌 （华南理工大学）

摘要：与经典Brown运动显著不同，分数Brown运动既不是半鞅也不是Markov过程，相应的动力学行为更为复杂。本报告将以分数Brown运动驱动的随机（偏）微分方程为对象，聚焦于随机稳定性、随机分岔、随机吸引子、随机不变流形等复杂动力学。

Averaging principles for controlled jump diffusions and associated nonlocal HJB equations

张琦 （清华大学）

摘要：Abstract: In this talk, we consider the multiscale problem of controlled jump diffusion with slow-fast timescales. The effective limit control problem is given via the averaging principles of controlled jump diffusions. Moreover, we investigate the main convergence theorem by two approaches: the tightness method of controlled jump diffusions and the perturbed test function method of associated nonlocal HJB equations. This is a joint work with Dr. Yanjie Zhang.

Smooth linearization for hyperbolic systems in local and global cases

张文萌（重庆师范大学）

摘要：In this talk, we first prove the random Belitskii’s C1 linearization theorem, whose original proof in the deterministic case has a gap.  In our novel proof, the Anosov splitting is an important tool. We show that it is C1 near the fixed point with a sharp non-resonant condition.  Then, we will further discussion the possibility to extend the local result on C1 linearization (as well as the C1 Anosov splitting) to global ones on  the torus to study the rigidity problem.

Well-posedness and dynamics of stochastic fractional Schrodinger equation on \mathbb{R}^n

张燕杰（郑州大学）

摘要：We establish the stochastic Strichartz estimate for the fractional Schr\"odinger  equation. With the help of the deterministic Strichartz estimates, we prove the existence and uniqueness of a global solution to the stochastic fractional nonlinear Schrodinger equation in L^2(\mathbb{R}^n) and $H^{\alpha}(\mathbb{R}^n)$ cases respectively. In addition, we also prove that this equation provides an infinite dimensional dynamical system that possesses a weak pullback mean random attractors in L^2(\mathbb{R}^n) .

非共形排斥子的维数理论

赵云 （苏州大学）

摘要：在本次报告中，我们将介绍非共形排斥子维数的研究进展，包括Bowen方程、维数的变分原理、维数逼近等。

Rotational entropy for random torus maps

朱玉峻（厦门大学）

摘要：In this talk, the rotational entropy is introduced for an i.i.d. random dynamical system on the torus. The formula of the rotational entropy is obtained for the system which satisfies certain assumptions, and the lower and upper bounds of the rotational entropy are given for more general systems. Several examples are presented to show that these results may not hold without the assumptions. This is a joint work with Jiang Weifeng and Lian Zhengxing.