Special Session 29: Stochastic Dynamical Systems

A mild rough Gronwall Lemma with applications to non-autonomous evolution equations

Alexandra Blessing University of Konstanz Germany Co-Author(s):    Mazyar Ghani Varzaneh, Tim Seitz

Abstract: We derive a Gronwall type inequality for mild solutions of non-autonomous parabolic rough partial differential equations (RPDEs). This inequality together with an analysis of the Cameron-Martin space associated to the noise, allows us to obtain the existence of moments of all order for the solution of the corresponding RPDE and its Jacobian when the random input is given by a Gaussian Volterra process. Applying further the multiplicative ergodic theorem, these integrable bounds entail the existence of Lyapunov exponents for RPDEs. We illustrate these results for stochastic partial differential equations with multiplicative boundary noise.

Most Probable Paths for McKean--Vlasov Systems: Deriving the Onsager--Machlup Functional via Euler-Type Classical SDEs

Ying Chao Xi`an Jiaotong University Peoples Rep of China Co-Author(s):

Abstract: The Onsager--Machlup functional is fundamental in characterizing fluctuations in nonequilibrium systems. For McKean--Vlasov stochastic differential equations, its derivation is challenging due to the distribution dependence in both drift and diffusion terms. In this talk, we introduce an Euler-type approximation scheme based on classical (distribution-independent) stochastic differential equations, each admitting an explicit Onsager--Machlup functional. By proving convergence to the McKean--Vlasov system, we rigorously obtain the corresponding functional in closed form. This constructive approach extends naturally to a broad class of distribution-dependent stochastic systems.

Random Attractors for McKean-Vlasov SDEs

Mengyu Cheng Dalian University of Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we focus on the existence of random attractors for McKean-Vlasov SDEs on a separable Hilbert space $H$. A significant challenge arises from the distribution-dependence of the coefficients, causing the lack of the stochastic flow property on $H$. To address this, we first transform the original equation into a system on the product space $H \times \mathcal{P}(H)$ and consider the existence of random attractors on this space. We then analyze cocycles associated with two parametric dynamical systems, define the corresponding pullback random attractor, and develop a general theory for the existence of random attractors for such cocycles. Finally, we apply our theoretical results to McKean-Vlasov stochastic ordinary differential equations, McKean-Vlasov stochastic reaction-diffusion equations, and McKean-Vlasov stochastic 2D Navier-Stokes equations. When the attractor reduces to a singleton $\mathcal{A}(\omega):=(\xi(\omega),\mu_{\infty})$, we show $\xi$ is the stationary solution for the decoupled stochastic partial differential equation, satisfying $\mathbb{P}\circ[\xi]^{-1}=\mu_{\infty}$. This talk is based on a joint work with Xianjin Cheng and Zhenxin Liu.

Amplitude Equations for SPDEs

Hongjun Gao Southeast University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we consider a class of stochastic partial differential equations (SPDEs) subject to a small deterministic perturbation that measures the distance to the change of stability, as well as a small random perturbation. Using multi-scale analysis, we derive amplitude equations for such SPDEs and then employ them to construct approximate solutions. We focus on the cases where the noise is given by standard Brownian motion or fractional Brownian motion.

Large Deviations for for Multiscale McKean-Vlasov Systems Driven by Fractional Brownian Motion

Jingyue Gao Southeast University Peoples Rep of China Co-Author(s):    Jingyue Gao; Hongjun Gao

Abstract: This work investigates the asymptotic behavior of multiscale McKean-Vlasov stochastic differential equations. The system consists of a slow component perturbed by a small fractional Brownian motion with Hurst index $H \in (1/2, 1)$ and a fast component driven by an independent standard Brownian motion. We establish the large deviation principle to characterize the probability of deviations from the averaged process. The main results are established via a weak convergence approach combined with the occupation measure method. The main challenge lies in addressing the inherent interplay between the distribution-dependent coefficients and the memory effects inherent in the fractional noise.

Random dynamical systems for McKean--Vlasov SDEs via rough path theory

Shanshan Hu TU Berlin Germany Co-Author(s):

Abstract: The existence of random dynamical systems for McKean--Vlasov SDEs is established. This is approached by considering the joint dynamics of the corresponding nonlinear Fokker-Planck equation governing the law of the system and the underlying stochastic differential equation (SDE) as a dynamical system on the product space $\RR^d \times \mathcal{P}(\RR^d)$. The proof relies on two main ingredients: At the level of the SDE, a pathwise rough path-based solution theory for SDEs with time-dependent coefficients is implemented, while at the level of the PDE a well-posedness theory is developed, for measurable solutions and allowing for degenerate diffusion coefficients. The results apply in particular to the so-called ensemble Kalman sampler (EKS), proving the existence of an associated RDS under some assumptions on the posterior, as well as to the Lagrangian formulation of the Landau equation with Maxwell molecules. As a by-product of the main results, the uniqueness of solutions to non-linear Fokker--Planck equations associated to the EKS is shown.

Local First Integrals and Poincar\`{e}-Type Nonintegrability for SDEs

Kaiyin Huang Sichuan Univeristy Peoples Rep of China Co-Author(s):

Abstract: This talk studies local first integrals for stochastic differential equations (SDEs). We define strong and weak local first integrals, and present necessary conditions for the existence of analytic first integrals by using resonance relations. The classical Poincar\`{e} nonintegrability theorem is extended to the stochastic framework.

A new branching diffusion solver for reaction-diffusion equations

Qiao Huang Southeast University Peoples Rep of China Co-Author(s):    Nicolas Privault

Abstract: Stochastic branching algorithms provide a useful alternative to grid-based schemes for the numerical solution of partial differential equations, particularly in high-dimensional settings. However, they require a strict control of the integrability of random functionals of branching processes in order to ensure the non-explosion of solutions. In this paper, we study the stability of a functional branching representation of PDE solutions by deriving sufficient criteria for the integrability of the multiplicative weighted progeny of stochastic branching processes. We also prove the uniqueness of mild solutions under uniform integrability assumptions on random functionals. This talk is based on joint work with N. Privault.

Inertial Manifolds Without Spectral Gap Conditions: Modified Navier-Stokes Equations

Xinhua Li School of Mathematics and Statistics, Lanzhou University Peoples Rep of China Co-Author(s):    Chunyou Sun

Abstract: Inertial manifold (IM) is a finite-dimensional invariant manifold that contains the global attractor and that attracts all the orbits at an exponential rate, and it is also a graph of some Lipschitz continuous functions. If a PDE possesses an IM, then its dynamical can be completely determined by a system of ODEs. Classical theory of IM required a so-called spectral gap condition for constructing an IM. In this talk, we introduce the method of spatial averaging which can construct IMs without spectral gap condition and consider the application for 2D modified Navier-Stokes equations (NSEs). An original motivation for the theory of IMs was treating the NSEs. Unfortunately, this problem is still open now. This talk will review key results concerning IMs for modified NSEs and present our recent contributions to this topic.

Dynamical order in noisy oscillators under unidirectional coupling on a line

Xiaofang Lin University of science and technology of China Peoples Rep of China Co-Author(s):    Bochun Chang; Yi Wang

Abstract: Dynamical order refers to the property that the global random attractor is a conal curve. It is established herein for noisy oscillator under unidirectional strong coupling on a line. The existence of one-dimensional global random attractor is obtained by identifying it as a manifold via stable foliation theory. We introduce the concept of random differentially positive systems to analyze its order structure. A key difficulty is that the evolution of a conal curve need not converge in general. To address this, we construct a suitably designed cone field that ensures uniform control over the growth of the evolution. By analyzing the evolution of a special conal curve, we prove that its evolution admits a sequential limit in the $C^{0}$-topology and that the global random attractor coincides with the limit, which is itself a closed conal curve, thereby establishing the dynamical order.

Limit Theorems for Inhomogeneous Markov Processes with Applications to SPDEs

Rongchang Liu Sichuan University Peoples Rep of China Co-Author(s):    Kening Lu; Bixiang Wang

Abstract: In this talk, we present recent progress on limit theorems for time-inhomogeneous Markov processes, including the strong law of large numbers, the central limit theorem, and both the Donsker-Varadhan and Freidlin-Wentzell large deviation principles. We assume that the time inhomogeneity is driven by a uniquely ergodic dynamical system and discuss applications to SPDEs.

Local stable and unstable sets for random dynamical systems

Xue Liu Southeast University Peoples Rep of China Co-Author(s):    Xiao Ma and Xiaomin Zhou

Abstract: In this talk, we study local stable and unstable sets for two-sided continuous bundle random dynamical systems with positive entropy, which serves as natural substitutes for invariant manifolds when $C^{1+\alpha}$ smoothness of the system is unavailable. For uniformly equicontinuous systems and ergodic invariant measures with positive entropy, we prove a lower bound for the Hausdorff dimension of local unstable sets in terms of the ratio of entropy to the maximal Lyapunov exponent. A symmetric statement for local stable sets is obtained by considering the reversed dynamics. Our results apply, in particular, to random dynamical systems generated by random differential equations with globally Lipschitz nonlinearities and to discrete-time systems arising from i.i.d. compositions of homeomorphisms drawn from a precompact subset of $C(X,X)$, where $X$ is a compact metric space. This is a joint work with Xiao Ma and Xiaomin Zhou.

An abstract criterion on the existence and global stability of stationary solutions for random dynamical systems and its applications

Xiang Lyu Shanghai Normal University Peoples Rep of China Co-Author(s):

Abstract: We prove a concise and easily verifiable criterion on the existence and global stability of stationary solutions for random dynamical systems (RDSs), which is very useful in a wide range of applications.

A Parameterization Method for Quasi-Periodic Systems with Noise: Computation of Random Invariant Tori

Pingyuan Wei Southeast University Peoples Rep of China Co-Author(s):    Lei Zhang

Abstract: In this talk, we focus on the study of normally hyperbolic invariant manifolds (NHIMs) for a class of quasi-periodically forced systems subject to additional stochastic noise. These systems can be interpreted as skew-product systems. The existence of NHIMs is established through the development of a parameterization method in the random setting, together with an application of the implicit function theorem in suitable Banach spaces. Based on this framework, we propose algorithms for computing the statistics of NHIMs as well as Lyapunov exponents. This talk is based on joint work with L. Zhang (DUT).