We study Dirichlet Green's functions associated with second-order parabolic systems with rapidly oscillating periodic coefficients that are symmetric and independent of time. For bounded C^1,1 domains, we obtain a sharp zeroth-order convergence estimate from the oscillating Green's function to its homogenized counterpart, with the optimal rate O(ε) and Gaussian off-diagonal decay. For bounded C^2,1 domains, we also prove a first-order expansion for the spatial gradient in terms of Dirichlet correctors, with an O(ε) error up to a logarithmic factor. In this time-independent symmetric setting, these results improve the convergence rates established by Geng in [Calc. Var. Partial Differ. Equ., 62(6), 2023] for parabolic systems with time-dependent periodic coefficient matrices.

In this paper, the stochastic Cucker-Smale (C-S) system with interaction potential and its corresponding McKean-Vlasov SDE are studied. Firstly, the existence and uniqueness of the considered equations are investigated by using the localizing approximation. By the corresponding Lyapunov functionals, we provide some flocking results for stochastic C-S system and McKean-Vlasov SDE. In addition, we provide some results on finite-time stability, finite-time mean-field limit, uniform-in-time stability, and uniform-in-time mean-field limit. Finally, several numerical simulations are employed to validate the effectiveness of the flocking results obtained by the C-S model.

This paper investigates the small-parameter limits of the generalized Langevin equation (GLE). It primarily covers two aspects: 1. We studied the white-noise limit of the GLE with state-dependent coefficients and a rich class of memory kernels that can be represented as the sum of infinite exponentials. The limiting equations are driven by white-noise. 2. To study the relationship between the white-noise limit and the small-mass limit, we investigated a one-dimensional GLE. First, we successively take the white-noise and small-mass limits for this equation, then reverse the order of limit-taking. The resulting limiting equations differ. Next, we simultaneously take the white-noise and small-mass limits with different convergence rates. When the convergence rates of the two parameters vary, the limiting equations are identical. The second part provides an intuitive guide for further study of the first part and its relationship to the small-mass limit. We used a technique of integrals of semimartingales established by Kurtz and Protter.

In this paper, we study a class of stochastic differential equations modeling diffusive phenomena with state-dependent and distribution-dependent friction. In the small-mass regime, the dynamics are shown to be governed by the Smoluchowski-Kramers approximation. We obtain the limiting equation and characterize the resulting additional drift terms via solutions of associated Lyapunov and Sylvester matrix equations. Moreover, we establish the rate of convergence and extend the model to incorporate more general interaction mechanisms and noise structures.

In this paper, we investigate the random dynamics of a system consisting of N interacting stochastic partial differential equations (SPDEs) with a mean-field interaction, where the interacting potential is Lipschitz continuous and odd. Leveraging the mean-field structure, we decompose the system into its ensemble average and the fluctuation component. A Lyapunov-Perron method is then employed to establish the existence of a finite-dimensional random invariant manifold for large interaction. Further we give a mean-field limit approximation of the reduced system on the random invariant manifold. Our result shows that random dynamics of the N interacting particle system is determined by that of the ensemble average part of the system for large interaction and the approximating system is deterministic as N ->infinity. At last, our results are clearly illustrated by one example.

This paper presents an effective approximation method for analyzing a stochastic system composed of interacting components for large . By assuming linear interactions and considering the ensemble average, we demonstrate that the system can be accurately approximated by a one-dimensional model. Furthermore, the approximation reveals the distinct influences of interaction and noise on the system's dynamics: While the interaction enhances metastability, the noise contributes to stabilizing the system.

An approximation is derived for a Langevin equation with distribution-dependent potential and state-dependent, randomly fast oscillation. By some estimates and a diffusion approximation the limiting equation is shown to be distribution-dependent stochastic differential equation (SDEs) driven by white noise.

We study fluctuation limit of a slow-fast system driven by α-stable Lévy noise with 1<α<2. The slow component is generated by an odd polynomial function f(y):=y^q, while in the fast component, the drift is g(y):=-|y|^psgn(y) for some p>0. Although the noise is given, the fluctuation limit is either a stable process or a Brownian motion, depending on both p and α. The critical line between stable limit and Brownian limit is q+1-p=α/2.

In this paper, we propose a spatiotemporal model incorporating memory-driven toxicant-taxis and nonlocal accumulation delay to investigate the interaction between the population and the toxicant in a polluted aquatic environment. Through linear stability theory and bifurcation analysis, we examine how the taxis coefficient, spatial memory delay, and average accumulation delay influence the stability of spatially homogeneous steady states and the formation of spatial patterns. Theoretical analysis reveals that the spatial memory delay does not affect the stability of the coexisting steady state, whereas the taxis coefficient and the average accumulation delay can destabilize it, triggering spatially heterogeneous patterns. We numerically validate these findings and illustrate how key toxicant-related parameters govern the distributions of the population and the toxicant. This study highlights the significant effects of memory-driven avoidance behavior and toxicant cumulative effects on shaping population distributions in polluted aquatic ecosystems.

The purpose of this paper is to establish the well-posedness of the stochastic Stefan problem on moving hypersurfaces. Through a specially designed transformation, it turns out we need to solve stochastic partial differential equations on a fixed hypersurface with a new kind of nonhomogeneous monotonicity involving a family of time-dependent operators. This new class of SPDEs is of independent interest and can also be applied to solve many other interesting models such as the stochastic p-Laplacian equations, stochastic Allen-Cahn equation and stochastic heat equations on time-dependent domains or hypersurfaces. (Monotone) Operator-valued calculus and geometric analysis of moving hypersurfaces play important roles in the study. Moreover, a forthcoming result on the well-posedness of stochastic 2D Navier-Stokes equation on moving domains is also based on our framework.

In this paper, the authors focus on the effective approximation (N →∞ then ε → 0) of stochastic interacting particle systems with fast regime-switching networks on digraph measures (DGMs). DGMs provide a robust approach to capturing sparse, intermediate, and dense network or graph interactions in the mean field, extending beyond traditional methods like graphons. The model can be used to simulate a vehicle’s trajectory under different traffic signal states. The main goals are to derive the simplified system (9) as ε → 0 and to capture a class of mean-field limits under the assumption that the switching process tends to a stationary state as time evolutions. Using the martingale method and validating the continuity of the underlying graph heterogeneity, the authors establish the convergence in law of (1) to a probability measure μ_t , which satisfies semi-linear Vlasov-Fokker-Planck equation.

We derive the amplitude equation for a system of N stochastically forced partial differential equations with cubic nonlinearity near a bifurcation point. The system features linear mean-field coupling, which introduces collective interactions among the components. By decomposing the dynamics into a slow mean-field component and N fast fluctuation modes, we employ an averaging method to derive the amplitude equation, valid both for large N and in the mean-field limit ( N ->infinity). Our analysis reveals how the interplay between mean-field interactions and additive noise influences the system's macroscopic behavior near criticality. These results provide a framework for understanding pattern formation and phase transitions in stochastically driven, collectively interacting systems.

We study the small mass limit for a class of systems described by McKean-Vlasov equations with different types of interaction potentials. Our aim is to identify the limiting equation and establish bounds on the error between solutions of the associated nonlinear kinetic Fokker-Planck equation and the limiting equation. We introduce an intermediate system through a coarse-graining map, enabling us to estimate the error between the spatial densities of the Vlasov-Fokker-Planck equation and the intermediate system in terms of the 2-Wasserstein distance. Subsequently, we derive an inequality for Wasserstein gradient flows by quantifying the error between the intermediate system and the corresponding limiting equation. This approach requires only the weak integrability of the interaction potentials, which quantifies the small mass limit of the nonlinear Fokker-Planck system with three different types of interaction potentials: bounded and Lipschitz interaction potential, singular interaction potential, and bounded interaction potential. Moreover, our approach includes examples of singular interactions such as the Riesz potential and the Coulomb potential.

We investigate approximation of random differential equations driven by semimartingales satisfying a singularly perturbed Langevin equation with scaled mixing random force. By a diffusion approximation approach, we explore the limit of the rough path lift of this semimartingale, and a universal limit theorem is applied to identify the limit of random differential equation. A structurally parallel proof also applies to establish an iterated weak invariance principle for the mixing random force, which is itself an independent interesting result. We find that, the limit of both of the second-level processes, have the form of iterated integral of Stratonovich form plus an anti-symmetric part which is proportional to the time increment.

We explore the limit of stochastic differential equations driven by some random processes satisfying singularly perturbed second order stochastic differential equations. The main tool we employ is the universal limit theorem in rough path theory. To this end, we lift the random process as a rough path in a natural manner. After suitable change-of-variable, the random process has a form of slow-fast system. Moment estimates of both the random process and its lift are given, followed by which, averaging technique and convergence theorem in rough path topology are used to identify the limit.

An L2(Rd)-valued stochastic N-interacting particles system with small mass is investigated. Mean field limit and the propagation of chaos are derived. Moreover the small mass limit of the solution is also built, which can be seen as a Smoluchowski-Kramers approximation on unbounded domain. Here a key step is the asymptotic compactness of the distribution of the solution, which is derived via a splitting technique of the domain Rd and some estimation of the solution for the mean field limit equation. We also show that the limits N -* infinity and F -* 0 commute. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The small mass limit is derived for a McKean-Vlasov equation subject to environmental noise with state-dependent friction. By applying the averaging approach to a non-autonomous stochastic slow-fast system with the microscopic and macroscopic scales, the convergence in distribution is obtained.

We explore the small mass limit of a stochastic wave equation (SWE) driven by cylindrical α-stable noise, where α∈ (1,2), and prove that it converges to a stochastic heat equation. We establish its well-posedness, and in particular, the càdlàg property, which is not trivial in the infinite dimensional case. Using a splitting technique, we decompose the velocity component into three parts, which gives convenience to the moment estimate. We show the tightness of solution of SWE by verifying the infinite dimensional version of Aldous condition. After these preparation, we pass the limit and derive the approximation equation.

The random homogenization for second order evolutionary equation with singular short-range correlated potential is derived. Comparing with the first order evolutionary equation, more difficulty need to be overcome in the moment estimation of the solution due to non-symmetrical semigroup of second order evolutionary equation. In our approach the solution is written out by the Duhamel's formula and the moment estimation of the solution is obtained by some analytic methods. Then by means of diagrammatic expansions and chaos expansions, the solution is shown to converge in distribution to the solution of stochastic partial differential equations (SPDEs) in Stratonovich form driven by spatial white noise. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.