arXiv subjects

Mengyu Cheng

Publications and source records attributed to Mengyu Cheng.

9 recordsLinked to original sources

arXiv2026

Averaging Principle and Pullback Attractor Convergence for McKean--Vlasov Stochastic Reaction--Diffusion Equations

We establish three averaging principles for distribution-dependent stochastic reaction--diffusion equations with rapidly oscillating coefficients on the torus $\mathbb T^d$, $d\le3$. First, solutions converge in mean square, uniformly on finite time intervals, to solutions of the averaged equation. Under a contraction condition, both the original and averaged equations admit unique bounded entire solutions whose mean-square distance vanishes uniformly for all $t\in\mathbb R$. At the level of probability laws, the original nonautonomous equation possesses a family of pullback attractors, whereas the averaged equation has a global attractor; the former converge upper-semicontinuously to the latter, uniformly over the coefficient hull. As an application, we present a class of stochastic reaction--diffusion models motivated by large-scale interacting systems.

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arXiv2025

Random Attractors for McKean-Vlasov SDEs

In this paper, we mainly focus on the existence of random attractors for McKean-Vlasov stochastic differential equations on a separable Hilbert space $H$. A significant challenge arises from the distribution-dependence of the coefficients, thereby causing the lack of the stochastic flow property on $H$. To address this issue, 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. Within this framework, we 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. In the case where the attractor reduces to a singleton set $\mathcal{A}(\omega):=(\xi(\omega),\mu_\infty)$, we show that $\xi$ corresponds to the stationary solution for the decoupled SPDE,satisfying $\mathbb{P}\circ[\xi]^{-1}=\mu_\infty$.

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arXiv2025

Strong averaging principle for nonautonomous multi-scale SPDEs with fully local monotone and almost periodic coefficients

In this paper, we consider a class of nonautonomous multi-scale stochastic partial differential equations with fully local monotone coefficients. By introducing the evolution system of measures for time-inhomogeneous Markov semigroups, we study the averaging principle for such kind of system. Specifically, we first prove the slow component in the multi-scale stochastic system converges strongly to the solution of an averaged equation, whose coefficients retain the dependence of the scaling parameter. Furthermore, if the coefficients satisfy uniformly almost periodic conditions, we establish that the slow component converges strongly to the solution of another averaged equation, whose coefficients are independent of the scaling parameter. The main contribution of this paper extends the basic nonautonomous framework investigated by Cheng and Liu in [11] to a fully coupled framework, as well as the autonomous framework explored by Liu et al. in [27] to the more general nonautonomous framework. Additionally, we improve the locally monotone coefficients discussed in [11,27] to the fully local monotone coefficients, thus our results can be applied to a wide range of cases in nonlinear nonautonomous stochastic partial differential equations, such as multi-scale stochastic Cahn-Hilliard-heat equation and multi-scale stochastic liquid-crystal-porous-media equation.

math.PR↗

arXiv2024

Averaging principle for SDEs with singular drifts driven by $α$-stable processes

In this paper, we investigate the convergence rate of the averaging principle for stochastic differential equations (SDEs) with $β$-Hölder drift driven by $α$-stable processes. More specifically, we first derive the Schauder estimate for nonlocal partial differential equations (PDEs) associated with the aforementioned SDEs, within the framework of Besov-Hölder spaces. Then we consider the case where $(α,β)\in(0,2)\times(1-\tfracα{2},1)$. Using the Schauder estimate, we establish the strong convergence rate for the averaging principle. In particular, under suitable conditions we obtain the optimal rate of strong convergence when $(α,β)\in(\tfrac{2}{3},1]\times(2-\tfrac{3α}{2},1)\cup(1,2)\times(\tfracα{2},1)$. Furthermore, when $(α,β)\in(0,1]\times(1-α,1-\tfracα{2}]\cup(1,2)\times(\tfrac{1-α}{2},1-\tfracα{2}]$, we show the convergence of the martingale solutions of original systems to that of the averaged equation. When $α\in(1,2)$, the drift can be a distribution.

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arXiv2023

Averaging principle and normal deviation for multi-scale SDEs with polynomial nonlinearity

We investigate three types of averaging principles and the normal deviation for multi-scale stochastic differential equations (in short, SDEs) with polynomial nonlinearity. More specifically, we first demonstrate the strong convergence of the solution of SDEs, which involves highly oscillating components and fast processes, to that of the averaged equation. Then we investigate the small fluctuations of the system around its average, and show that the normalized difference weakly converges to an Ornstein-Uhlenbeck type process, which can be viewed as a functional central limit theorem. Additionally, we show that the attractor of the original system tends to that of the averaged equation in probability measure space as the time scale $\varepsilon$ goes to zero. Finally, we establish the second Bogolyubov theorem; that is to say, we prove that there exists a quasi-periodic solution in a neighborhood of the stationary solution of the averaged equation when the $\varepsilon$ is small.

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arXiv2022

Averaging principle for stochastic complex Ginzburg-Landau equations

Averaging principle is an effective method for investigating dynamical systems with highly oscillating components. In this paper, we study three types of averaging principle for stochastic complex Ginzburg-Landau equations. Firstly, we prove that the solution of the original equation converges to that of the averaged equation on finite intervals as the time scale $\varepsilon$ goes to zero when the initial data are the same. Secondly, we show that there exists a unique recurrent solution (in particular, periodic, almost periodic, almost automorphic, etc.) to the original equation in a neighborhood of the stationary solution of the averaged equation when the time scale is small. Finally, we establish the global averaging principle in weak sense, i.e. we show that the attractor of original system tends to that of the averaged equation in probability measure space as $\varepsilon$ goes to zero.

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arXiv2022

Strong and weak convergence for averaging principle of DDSDE with singular drift

In this paper, we study the averaging principle for distribution dependent stochastic differential equations with drift in localized $L^p$ spaces. Using Zvonkin's transformation and estimates for solutions to Kolmogorov equations, we prove that the solutions of the original system strongly and weakly converge to the solution of the averaged system as the time scale $\eps$ goes to zero. Moreover, we obtain rates of the strong and weak convergence that depend on $p$ respectively.

math.PR↗

arXiv2022

The second Bogolyubov theorem and global averaging principle for SPDEs with monotone coefficients

In this paper, we establish the second Bogolyubov theorem and global averaging principle for stochastic partial differential equations (in short, SPDEs) with monotone coefficients. Firstly, we prove that there exists a unique $L^{2}$-bounded solution to SPDEs with monotone coefficients and this bounded solution is globally asymptotically stable in square-mean sense. Then we show that the $L^{2}$-bounded solution possesses the same recurrent properties (e.g. periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, Levitan almost periodic, etc.) in distribution sense as the coefficients. Thirdly, we prove that the recurrent solution of the original equation converges to the stationary solution of averaged equation under the compact-open topology as the time scale goes to zero--in other words, there exists a unique recurrent solution to the original equation in a neighborhood of the stationary solution of averaged equation when the time scale is small. Finally, we establish the global averaging principle in weak sense, i.e. we show that the attractor of original system tends to that of the averaged equation in probability measure space as the time scale goes to zero. For illustration of our results, we give two applications, including stochastic reaction diffusion equations and stochastic generalized porous media equations.

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arXiv2019

Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients

In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential equations with monotone coefficients. Firstly, we establish the continuous dependence on initial values and coefficients for solutions. Secondly, we prove the existence of recurrent solutions, which include periodic, almost periodic and almost automorphic solutions. Then we show that these recurrent solutions are globally asymptotically stable in square-mean sense. Finally, for illustration of our results we give two applications, i.e. stochastic reaction diffusion equations and stochastic porous media equations.

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