Special Session 176: Non-local Stochastic Evolutionary Systems: Theory and Applications

Global Solvability for Stochastic Superconducting Model

Nikolai V Chemetov University of Sao Paulo Brazil Co-Author(s):

Abstract: In this work we study the stochastic (Stratonovich) superconducting model with an infinite dimensional multiplicative noise in a bounded domain {% \ $\mathcal{O}$ in $\mathbb{R}^{2}$}. The evolution system on a time interval $(0,T)$ reads for a.s. in $\Omega $ as \begin{equation*} \left\{ \begin{array}{l} du= \mathrm{div}(g(u)\ \nabla h)\,dt+\sigma (u)\circ d{\mathcal{W}}_{t}, \ \ -\Delta h+h=u,% \end{array}% \right. \quad\quad \text{for}\quad {(t,\mathbf{x})\in }(0,T)\times \mathcal{O} \end{equation*}% where {$u=u(t,\mathbf{x})$} denotes the cell density and $g(u)=|u|;$ ${% \mathcal{W}}_{t}$ is a Wiener process given on the probability space $(\Omega ,\mathcal{F},P)$ and $\sigma =\sigma (u)$ is the diffusion coefficient. \ {% This system is closed by the boundary condition \begin{equation*} h=a \quad \quad \quad \text{on}\quad (0,T)\times \partial {% \mathcal{O}} \end{equation*}% on the boundary }$\partial $$\mathcal{O}$\ of the domain $\mathcal{O},$ {\, the influx boundary condition }% \begin{equation*} u=b \quad \text{on}\quad (0,T)\times \partial {\mathcal{O}}^{% \mathbf{-}} \end{equation*}% and the initial condition% \begin{equation*} u=u_{0}\quad \quad \quad \text{in}\quad \mathcal{O} \quad \text{at}\quad t=0, \end{equation*}% where \begin{equation*} \partial {\mathcal{O}}^{\mathbf{-}}=\left\{ \mathbf{x}\in \partial {\mathcal{% O}}:\quad g^{\prime }(u)\left( \nabla h\cdot \mathbf{n}\right) (t,\mathbf{x}% )

A Dimension-Free Limiting Formula for Negative Sobolev Norms

Huaiqian Li Tianjin University Peoples Rep of China Co-Author(s):

Abstract: We establish a dimension-free limiting formula for the norms of negative Sobolev spaces. These spaces play a fundamental role in fluid mechanics, and their norms are conveniently characterized via the heat kernel. Our main result captures a sharp nonlocal-to-local transition: as the fractional order tends to zero, negative Sobolev norm converges to a $L^p$ norm. Our method is based on basic properties of the heat kernel and $L^p$-norm approximation techniques.

Eddy viscosity by Levy transport noise

Dejun Luo Academy of Mathematics and Systems Science, Chinese Academy of Sciehces Peoples Rep of China Co-Author(s):

Abstract: We consider stochastic 2D Euler equations with $L^2$-initial vorticity and driven by L\`evy transport noise in the Marcus sense. Under a suitable scaling limit of the noises, we prove that the weak solutions converge weakly to the unique solution of the deterministic 2D Navier-Stokes equation. This shows that small scale jump noises generate eddy viscosity, extending the recent studies on It\^o-Stratonovich diffusion limit to discontinuous setting. This is a joint work with Dr. Feifan Teng.

Convergence and Fluctuations of Sginular Interacting Particle Systems with Non-Markov Noise

Avi Mayorcas University of Bath England Co-Author(s):

Abstract: Systems of interacting particles are ubiquitous in physics, biology, chemistry, computer science and the social sciences. In many applications, the desired interaction potential is a highly singular function or distribution, making well-posedness, mean field convergence, and precise fluctuations challenging to obtain. A particularly relevant example are point vortex models of fluids where both singular interactions and non-Gaussian correlations are practically relevant. In this talk I will present recent results, obtained with L. Galeati and K. Le as well as ongoing work with J. Weinberger, on interacting particle systems driven by i.i.d. fractional Brownian motions (fBm), subject to irregular, possibly distributional, pairwise interactions. We show quantitative propagation of chaos and Gaussian fluctuations for these systems, where the singularity of the interaction may be chosen arbitrarily severe provided the noise fluctuates sufficiently fast (in a sense to be made formal). Our proofs rely on a combination of Sznitman`s direct comparison argument with stochastic sewing techniques and U-statistics. Time permitting I will also present complimentary results on multiplicative equations driven by fBm with singular volatitlity and ergodic theory for fBm driven coercive SDEs with singular drift.

Ergodicity of stochastic PDEs on unbounded domains

Vahagn Nersesyan NYU Shanghai Peoples Rep of China Co-Author(s):

Abstract: In the last two decades, there has been significant progress in the understanding of ergodic properties of white-forced dissipative PDEs. The previous studies mostly focus on equations posed on bounded domains since they rely on different compactness properties and the discreteness of the spectrum of the Laplacian. In this talk, we will review recent results that make it possible to treat unbounded domains.

Intrinsic vs. Extrinsic Noise in Transient Dynamics: A Quasi-Stationary Approach

Weiwei Qi Academy of Mathematics and Systems Science, CAS Peoples Rep of China Co-Author(s):

Abstract: Transient dynamics--long-lasting but ultimately finite-time behaviors--are ubiquitous in complex stochastic systems. Understanding their underlying mechanisms remains a fundamental challenge. In this talk, we investigate randomly perturbed processes arising in chemical reaction networks and population dynamics, where extinction is inevitable but preceded by long-lived persistence. Using quasi-stationary distributions (QSDs), we characterize these transient dynamics and analyze their asymptotics in the vanishing-noise regime. We show that intrinsic and extrinsic noise can induce fundamentally different transient dynamics, leading to distinct persistence and extinction patterns. We conclude with a discussion of broader implications and open problems.

On the stability of stochastic processes with regime-switching

Jinghai Shao Tianjin University Peoples Rep of China Co-Author(s):

Abstract: The stochastic processes with regime-switching are extensively used for modeling the realistic systems in control engineering, which experience abrupt changes in structure or parameters. This talk concerns the long time behavior of regime-switching systems, focusing on the impact of infinitely countable switching states and the delayed observations.

Stochastic Euler Equations with Pseudo-differential Noise

Hao Tang Tianjin University Peoples Rep of China Co-Author(s):

Abstract: We study stochastic Euler equations in both compressible and incompressible regimes, on the whole space and on the torus, driven by genuinely mixed multiplicative noise comprising a continuous component (in both the Stratonovich and It\^o senses) and a discontinuous component (in the Marcus sense). The Stratonovich/Marcus noise amplitudes are (nonlocal) pseudo-differential operators that include, as special cases, the classical transport operator. Within this setting, we develop a local-in-time theory of classical solutions for both regimes, establishing existence, uniqueness, and a blow-up criterion. The inclusion of discontinuous pseudo-differential Marcus noise is novel and necessitates additional analytical techniques to control the interaction between jumps and nonlocal operators. In the compressible barotropic case, we consider a broad class of physically relevant pressure laws extending far beyond the polytropic $\gamma$-law. This class includes piecewise-defined $\gamma$-laws, (piecewise-defined) Chaplygin-type laws, the pressure of white dwarf stars, and other astrophysically relevant regimes. These pressure laws have not been analyzed in the literature on stochastic compressible fluids, even under purely It\^o-type forcing. For the incompressible damped equations, we specify a hierarchy of damping-noise conditions of increasing strength that yield global-in-time existence, uniform-in-time bounds, and decay estimates, respectively. To analyze invariant measures, we formulate an abstract singular stochastic evolution system that captures Euler-specific features, notably the mismatch between the topology in which solutions are constructed and the topology in which the Feller property holds. We extend the Krylov--Bogoliubov argument to accommodate this mismatch, establishing existence and uniqueness of invariant measures for the stochastic incompressible damped Euler equations. This provides the first positive answer to a strengthened version of Shirikyan`s problem.

Statistically stationary solutions to the stochastic compressible Euler equations with linear damping

Krutika Tawri University of Washington USA Co-Author(s):    Jeffrey Kuan, Konstantina Trivisa

Abstract: In this talk, we will discuss the existence of statistically stationary solutions to the linearly damped stochastic compressible Euler equations in one spatial dimension. The system of isentropic compressible Euler equations, describing the evolution of the density and momentum of a compressible fluid, is driven by a multiplicative noise. Additionally, the momentum equation consists of a small linear damping term. The power law for the pressure is given in terms of the density. We show the existence of statistically stationary weak martingale entropy solutions to these equations for any adiabatic constant and any damping coefficient, namely stochastic solutions for which the law of the solution is constant in time. To establish this result, we consider an approximate system for which we show the existence of an invariant measure, and then pass to the limit in the approximation parameters in order to obtain a limiting statistically stationary solution. We develop new techniques for obtaining uniform-in-time estimates for entropies of all orders, and study invariant regions of the approximate system. Such a result is a significant step in understanding the long-time statistics of stochastically perturbed compressible inviscid fluids.

Quantitative Convergence and Gaussian Fluctuations for Sequential Interacting Diffusions via Incremental Relative Entropy

Zhenfu Wang Peking University Peoples Rep of China Co-Author(s):    Xianliang Zhao

Abstract: We study a sequential system of interacting diffusions in which particle $i$ interacts only with its predecessors through the empirical measure $\mu_t^{i-1}$, yielding a directed, non-exchangeable mean-field approximation of a McKean--Vlasov diffusion. Under bounded coefficients and a non-degenerate constant diffusion, we prove the sharp decay of incremental path-space relative entropies, $$ R_i(T):=H\!\left(P^{1:i}_{[0,T]}\,\middle|\,P^{1:i-1}_{[0,T]}\otimes \bar P_{[0,T]}\right) \ \lesssim\ \frac{1}{i-1}, \qquad i\ge2, $$ where $P^{1:i}_{[0,T]}$ is the law of the first $i$ particle paths and $\bar P_{[0,T]}$ the McKean--Vlasov path law. Summing the increments yields the global estimate $$ H\left(P^{1:N}_{[0,T]}\,\middle|\,\bar P_{[0,T]}^{\otimes N}\right)\ \lesssim\ \log N, $$ together with quantitative decoupling bounds for tail blocks. As a consequence, the empirical measure converges to the McKean--Vlasov equation in negative Sobolev topologies at the canonical $N^{-1/2}$ scale. We also establish a Gaussian fluctuation limit for the fluctuation measure, where the sequential architecture produces an explicit feedback correction in the limiting linear SPDE. Our proofs rely on a Girsanov representation, a martingale-difference replacement of predecessor empirical measures by average conditional measures, and an upper-envelope argument.

Mean Field Control with Poissonian Common Noise: A Pathwise Compactification Approach

Xiaoli Wei Harbin Institute of Technology Peoples Rep of China Co-Author(s):

Abstract: This paper contributes to the compactification approach to study mean-field control problems with Poissonian common noise. To overcome the lack of compactness and continuity issues caused by common noise, we exploit the point process representation of the Poisson random measure with finite intensity and propose a pathwise formulation in a two step procedure by freezing a sample path of the common noise. In the first step, we establish the existence of the optimal relaxed control in the pathwise formulation as if common noise is absent, but with finite deterministic jumping times. The second step plays the key role in our approach, which is to aggregate the optimal solutions in the pathwise formulation over all sample paths of common noise and show that it yields an optimal solution in the original model. To this end, with the help of concatenation techniques, we first develop a pathwise superposition principle in the model with deterministic jumping times, drawing a relationship between the pathwise relaxed control problem and the pathwise measure-valued control problem. As a result, we can further bridge the equivalence among different problem formulations and verify that the constructed solution under aggregation is indeed optimal in the original problem.