Xiaobin Sun, Huilian Xia, Yingchao Xie, Xingcheng Zhou
In this paper, the strong averaging principle is researched for a class of Hölder continuous drift slow-fast SPDEs with $α$-stable process by the Zvonkin's transformation and the classical Khasminkii's time discretization method. As applications, an example is also provided to explain our result.
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Our first result is a stochastic sewing lemma with quantitative estimates for mild incremental processes, with which we study SPDEs driven by fractional Brownian motions in a random environment. We obtain uniform $L^p$-bounds. Our second result is a fractional averaging principle admitting non-stationary fast environments. As an application, we prove a fractional averaging principle for SPDEs.
In this paper, we first study the well-posedness of a class of McKean-Vlasov stochastic partial differential equations driven by cylindrical $α$-stable process, where $α\in(1,2)$. Then by the method of the Khasminskii's time discretization, we prove the averaging principle of a class of multiscale McKean-Vlasov stochastic partial differential equations driven by cylindrical $α$-stable processes. Meanwhile, we obtain a specific strong convergence rate.
In this paper, we aim to study the asymptotic behaviour for a class of McKean-Vlasov stochastic partial differential equations with slow and fast time-scales. Using the variational approach and classical Khasminskii time discretization, we show that the slow component strongly converges to the solution of the associated averaged equation. In particular, the corresponding convergence rates are also obtained. The main results can be applied to demonstrate the averaging principle for various McKean-Vlasov nonlinear SPDEs such as stochastic porous media type equation, stochastic $p$-Laplace type e…
In this paper, we study a class of slow-fast stochastic partial differential equations with multiplicative Wiener noise. Under some appropriate conditions, we prove the slow component converges to the solution of the corresponding averaged equation with optimal orders 1/2 and 1 in the strong and weak sense respectively. The main technique is based on the Poisson equation.
In this paper, the averaging principle is studied for a class of multiscale stochastic partial differential equations driven by $α$-stable process, where $α\in(1,2)$. Using the technique of Poisson equation, the orders of strong and weak convergence are given $1-1/α$ and $1-r$ for any $r\in (0,1)$ respectively. The main results extend Wiener noise considered by Bréhier in [6] and Ge et al. in [17] to $α$-stable process, and the finite dimensional case considered by Sun et al. in [39] to the infinite dimensional case.
Strong averaging principle for nonautonomous slow-fast SPDEs driven by $α$-stable processes
本文针对由$\alpha$-稳定过程驱动的非自治慢快随机偏微分方程，通过演化测度系统建立平均原理，证明慢分量在$L^p$意义下强收敛，并处理了缺乏有限二阶矩的技术难题。
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…
Orders of convergence in the averaging principle for SPDEs: the case of a stochastically forced slow component
这篇文章致力于分析半线性，抛物，随机偏微分方程，慢和快时间尺度。渐近地，平均原理成立：慢分量收敛到另一个半线性，抛物线，SPDE的解决方案，其中非线性平均相对于快速过程的不变分布。我们展示了收敛的顺序，在强和弱的意义上，在两个相关的情况下，取决于快速过程的空间规律性和慢方程中的维纳噪声的协方差。在非常规则的情况下，强和弱阶等于$\frac12$和$1$。在不太规则的情况下，弱序也是强序的两倍。这项研究扩展了以前的弱收敛速度的结果，无论是没有随机强迫项包括在慢方程，或噪声的协方差是非常经常的。一个有效的数值方案，基于非均匀多尺度方法进行了简要的讨论。
This paper is devoted to proving the strong averaging principle for slow-fast stochastic partial differential equations with locally monotone coefficients, where the slow component is a stochastic partial differential equations with locally monotone coefficients and the fast component is a stochastic partial differential equations (SPDEs) with strongly monotone coefficients. The result is applicable to a large class of examples, such as the stochastic porous medium equation, the stochastic $p$-Laplace equation, the stochastic Burgers type equation and the stochastic 2D Navier-Stokes equation, …
In this work, we study a class of non-autonomous two-time-scale stochastic reaction-diffusion equations driven by Poisson random measures, in which the coefficients satisfy the polynomial growth condition and local Lipschitz condition. First, the existence and uniqueness of the mild solution are proved by constructing auxiliary equations and using the technique of stopping time. Then, consider that the time dependent of the coefficients, the averaged equation is redefined by studying the existence of time-dependent evolution family of measures associated with the frozen fast equation. Further,…
Strong and weak order in averaging for SPDEs
针对具有慢速和快速时间尺度的抛物型随机发展方程系统，证明了平均化结果，并给出了强收敛阶1/2和弱收敛阶1的显式误差界。
Averaging principle for a class of stochastic reaction-diffusion equations
考虑随机反应扩散方程的平均原理。在某些假设下，提供了一个唯一的不变测度的快速运动与冻结的慢分量的存在，我们计算限制慢运动。对希尔伯特空间中Kolmogorov方程的可解性的研究和解的正则性分析，使得经典方法在SPDE的情况下推广到有限维问题。
This paper is devoted to a system of stochastic partial differential equations (SPDEs) that have a slow component driven by fractional Brownian motion (fBm) with the Hurst parameter $H >1/2$ and a fast component driven by fast-varying diffusion. It improves previous work in two aspects: Firstly, using a stopping time technique and an approximation of the fBm, we prove an existence and uniqueness theorem for a class of mixed SPDEs driven by both fBm and Brownian motion; Secondly, an averaging principle in the mean square sense for SPDEs driven by fBm subject to an additional fast-varying diffus…
Strong averaging principle for nonautonomous multi-scale SPDEs with fully local monotone and almost periodic coefficients
针对带完全局部单调概周期系数的非自治多尺度SPDE，引入测度演化系统证明慢分量的强平均原理，拓展既有框架并放宽系数条件，可应用于多类非线性随机偏微分方程。
In order to characterize the fluctuation between the ergodic limit and the time-averaging estimator of a full discretization in a quantitative way, we establish a central limit theorem for the full discretization of the parabolic stochastic partial differential equation. The theorem shows that the normalized time-averaging estimator converges to a normal distribution with the variance being the same as that of the continuous case, where the scale used for the normalization corresponds to the temporal strong convergence order of the considered full discretization. A key ingredient in the proof …
Points of slow growth for parabolic SPDEs
本文研究了抛物型随机偏微分方程在时间零附近的慢增长点，证明了存在随机点使得解的波动具有$t^{1/4}$阶量级。
We study an asymptotic preserving scheme for the temporal discretization of a system of parabolic semilinear SPDEs with two time scales. Owing to the averaging principle, when the time scale separation $ε$ vanishes, the slow component converges to the solution of a limiting evolution equation, which is captured when the time-step size $Δt$ vanishes by a limiting scheme. The objective of this work is to prove weak error estimates which are uniform with respect to $ε$, in terms of $Δt$: the scheme satisfies a uniform accuracy property. This is a non trivial generalization of a recent article in …
We study stochastic partial differential equations (SPDEs) with potentially very rough fractional noise with Hurst parameter $H\in(0,1)$. Close to a change of stability measured with a small parameter $\varepsilon$, we rely on the natural separation of time-scales and establish a simplified description of the essential dynamics. We prove that up to an error term bounded by a power of $\varepsilon$ depending on the Hurst parameter we can approximate the solution of the SPDE in first order by an SDE, the so called amplitude equation, and in second order by a fast infinite dimensional Ornstein-Uh…
By using the technique of the Zvonkin's transformation and the classical Khasminkii's time discretization method, we prove the averaging principle for slow-fast stochastic partial differential equations with bounded and Hölder continuous drift coefficients. An example is also provided to explain our result.
Concentration estimates for SPDEs driven by fractional Brownian motion
本文针对柱形分数布朗运动驱动的慢时变SPDE，避开鞅方法，利用高斯过程上确界尾概率最优估计，得到Hurst指数H>1/4时解的样本路径浓度估计，还附带得到任意H∈(0,1)下一维分数SDE的浓度估计。
Concentration estimates for slowly time-dependent singular SPDEs on the two-dimensional torus
本文研究二维环面上受弱时空白噪声驱动的慢时变奇异随机偏微分方程，在 Wick 重整化下建立样本路径在稳定平衡分支附近的集中估计，并讨论叉形分岔情形，从而将有限维和一维环面的已有结果推广到二维环面。
This paper is devoted to studying the averaging principle for stochastic differential equations with slow and fast time-scales, where the drift coefficients satisfy local Lipschitz conditions with respect to the slow and fast variables, and the coefficients in the slow equation depend on time $t$ and $ω$. Making use of the techniques of time discretization and truncation, we prove that the slow component strongly converges to the solution of the corresponding averaged equation.
Dirichlet-Neumann Averaging: The DNA of Efficient Gaussian Process Simulation
本研究提出基于Dirichlet-Neumann平均的高效高斯过程/随机场采样方法，结合离散余弦与正弦变换实现快速计算，同时建立其与SPDE方法的关联，可指导边界条件选择并在无域扩展下得到各向同性样本。
Averaging principle for non autonomous slow-fast systems of stochastic RDEs: the almost periodic case
研究了一类慢-快随机反应扩散方程系统的平均原理的有效性。这里我们假设快方程的系数依赖于时间，因此，用快方程的不变测度表示的平均原理的经典公式不再适用。作为一种替代方案，我们引入了与快速方程相关的随时间演化的措施。在快方程中的系数是概周期的假设下，测度族的演化是概周期的。这允许识别适当的平均方程并证明平均极限的有效性。
In this paper, we consider a class of slow-fast systems of stochastic partial differential equations where the nonlinearity in the slow equation is not continuous and unbounded. We first provide conditions that ensure the existence of a martingale solution. Then we prove that the laws of the slow motions are tight, and any of their limiting points is a martingale solution for a suitable averaged equation. Our results apply to systems of stochastic reaction-diffusion equations where the reaction term in the slow equation is only continuous and has polynomial growth.
Averaging principle for slow-fast fractional stochastic differential equations
该研究证明了慢-快耦合的分数阶随机微分系统的平均原理，确立了慢分量在时间尺度分离趋于零时均方收敛于自治平均方程解的结论，并在快分量不依赖慢分量时给出了依赖分数阶导数阶数的收敛速率。
Averaging principle for semilinear slow-fast rough partial differential equations
本文针对由有限维粗糙乘性噪声驱动的半线性慢-快偏微分方程，利用受控粗糙路径理论与 Khasminskii 时间离散化方法，证明了慢分量在 Hölder 拓扑下强收敛到平均化方程的解。
Averaging principle for one dimensional stochastic Burgers equation
本文研究了具有慢时标和快时标的一维随机Burgers方程的平均原理。在适当的条件下，我们证明了慢分量强收敛于相应的平均方程的解。同时，当慢分量方程中不存在噪声时，我们还证明了对任意$0<r<1$，慢分量弱收敛于相应的平均方程的解，其收敛阶为$1-r$.
In the present work, we establish the approximation of nonlinear stochastic partial differential equation (SPDE) driven by cylindrical α-stable Lévy processes via modulation or amplitude equations. We study SPDEs with a cubic nonlinearity, where the deterministic equation is close to a change of stability of the trivial solution. The natural separation of time-scales close to this bifurcation allows us to obtain an amplitude equation describing the essential dynamics of the bifurcating pattern, thus reducing the original infinite dimensional dynamics to a simpler finite-dimensional effective d…
The order of convergence in the averaging principle for slow-fast systems of stochastic evolution equations in Hilbert spaces
本文研究 Hilbert 空间中带一般加性 Wiener 噪声的随机发展方程快慢系统，证明慢分量向平均方程的强收敛阶为最优的 1/2，并应用于时空白噪声驱动的随机反应扩散系统。
Effective dynamics of interfaces for nonlinear SPDEs driven by multiplicative white noise
本文研究乘性白噪声驱动的无穷维SPDE界面动力学，利用有限维SDE推导慢流形上的有效方程，验证SPDE与耦合系统在小随机扰动下的等价性，并在四类随机方程中展示应用。
This work studies the averaging principle for a fully coupled two time-scale system, whose slow process is a diffusion process and fast process is a purely jumping process on an infinitely countable state space. The ergodicity of the fast process has important impact on the limit system and the averaging principle. We showed that under strongly ergodic condition, the limit system admits a unique solution, and the slow process converges in the L1-norm to the limit system. However, under certain weaker ergodicity condition, the limit system admits a solution, but not necessarily unique, and the …
MDP for integral functionals of fast and slow processes with averaging
本文建立了向量值随机过程族$(X^ε,Y^ε),ε\to 0$的大偏差原理，定义为$$ X^ε_t=\frac{1}{ε^κ}\int_0^t H(ξ^ε_s,Y^ε_s)ds, dY^ε_t=F(ξ^ε_t,Y^ε_t)dt+ Dε^{1/2-κ}G(ξ^ε_t,Y^ε_t)dW_t,$$，其中$W_t$是Wiener过程，$ξ^ε_t$是快速遍历扩散过程.我们证明了在$κ<{1/2}$或更小和Veretennikov-Khasminskii型快速扩散条件下，LDP满足Freidlin-Wentzell型速率函数.
Strong convergence rate in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales
本文研究了具有两个时间尺度的随机双曲-抛物型方程的平均原理，其中快、慢分量均受乘性噪声扰动。特别是，我们证明了慢分量的平均动力学的强收敛速度是为了$1/2$，这显着提高了我们以前的工作中建立的顺序$1/4$。
In this paper, we study a system of stochastic partial differential equations with slow and fast time-scales, where the slow component is a stochastic real Ginzburg-Landau equation and the fast component is a stochastic reaction-diffusion equation, the system is driven by $α$-stable process with $α\in (1,2)$. Using the classical Khasminskii approach based on time discretization and the techniques of stopping times, we show that the slow component strong converges to the solution of the corresponding averaged equation under some suitable conditions.
We consider slow / fast systems where the slow system is driven by fractional Brownian motion with Hurst parameter $H>{1\over 2}$. We show that unlike in the case $H={1\over 2}$, convergence to the averaged solution takes place in probability and the limiting process solves the 'naïvely' averaged equation. Our proof strongly relies on the recently obtained stochastic sewing lemma.
Averaging principle for two dimensional stochastic Navier-Stokes equations
分别对二维随机Navier-斯托克斯方程和二维随机反应扩散方程的慢分量和快分量建立了平均原理。利用基于时间离散的经典Khasminskii方法，在适当的条件下证明了慢分量强收敛于相应的平均方程的解.同时，采用了一些强有力的技巧来克服非线性项带来的困难，释放初值的规律性。
Averaging Principle for Ordinary Differential Equations in a Multiscale Random Environment
该研究针对快慢随机系统，在适当条件下证明向量场依赖快慢过程的微分方程解依测度收敛到仅依赖慢过程的平均系统，将经典平均理论推广到受多分量快随机环境影响的系统。
We study a large deviation principle for a system of stochastic reaction--diffusion equations (SRDEs) with a separation of fast and slow components and small noise in the slow component. The derivation of the large deviation principle is based on the weak convergence method in infinite dimensions, which results in studying averaging for controlled SRDEs. By appropriate choice of the parameters, the fast process and the associated control that arises from the weak convergence method decouple from each other. We show that in this decoupling case one can use the weak convergence method to charact…
Weak order in averaging principle for two-time-scale stochastic partial differential equations
研究了有界区间$[0, l]$上一类两时间尺度随机偏微分方程的平均原理，其中快、慢分量均受加性噪声直接扰动.在漂移系数的正则条件下，证明了对于任意小的$\varepsilon>0$，慢变量对平均动力学的弱收敛速度是$1-\varepsilon$阶.证明是基于一个渐近展开的多时间尺度系统相关的Kolmogorov方程的解决方案。
Averaging principles for time-inhomogeneous multi-scale SDEs with partially dissipative coefficients
本文研究时间非齐次多尺度随机微分方程的平均化原理，针对快速分量漂移仅部分耗散的情形，分别建立了慢分量强收敛与弱收敛结果，并基于演化测度系的存在唯一性证明。
Scaling Laws and Warning Signs for Bifurcations of SPDEs
临界转变（或临界点）是在许多动力系统中观察到的剧烈的突然变化。大量的临界跃迁与系统相关联，它们缓慢地向歧点漂移。在随机常微分方程（SODEs）的背景下，有一些关于变点前方差增长和自相关的结果，这些结果可以作为应用中可能的警告信号。一个类似的理论最近已经发展在最简单的设置也为随机偏微分方程（SPDE）的自伴算子的漂移项。这种设置导致真实的离散谱和协方差算子通过一定的标度律的增长。在本文中，我们进一步发展了这一理论。我们涵盖了复本征值、简并本征值以及连续谱的情形。这为SPDE分叉的警告信号的大多数实际应用提供了相当全面的理论。
This paper is devoted to studying the averaging principle for fast-slow system of rough differential equations driven by mixed fractional Brownian rough path. The fast component is driven by Brownian motion, while the slow component is driven by fractional Brownian motion with Hurst index $H ~(1/3 < H\leq 1/2)$. Combining the fractional calculus approach to rough path theory and Khasminskii's classical time discretization method, we prove that the slow component strongly converges to the solution of the corresponding averaged equation in the $L^1$-sense. The averaging principle for a fast-slow…
Averaging principle for rough slow-fast systems of level 3
本文针对由3阶随机粗糙路径驱动的粗糙慢-快系统，提出了相应的系统模型，并证明了Khas'minskii型的强平均原理。
Averaging principle for jump processes depending on fast ergodic dynamics
本文研究慢-快随机过程，证明当快分量遍历动力学加速时，慢跳跃过程收敛到自治纯跳跃过程，并应用于分支过程与流行病模型的收敛性证明。
On large deviations in the averaging principle for SDE's with a "full dependence", correction
在快分量的系数依赖于慢分量的情况下，建立了含平均的随机微分方程的大偏差原理。
Coadjoint averaging and cross-scale fluxes in a fast-slow stochastic Euler-Arnold system on $SU(N)$
研究 Zeitlin 的 SU(N) 近似随机版中能量和涡度相关跨尺度通量，引入快速-慢速随机框架，表明余伴随对称性抑制平均非线性通量，其打破可选择跨尺度转移方向。
Averaging principle for slow-fast systems of PDEs with rough drivers
本文针对粗糙偏微分方程慢快系统，利用受控粗糙路径框架与时间离散化，证明了当时间尺度参数趋于零时慢分量强收敛于平均化系统的解。
We focus on fast-slow systems involving both fractional Brownian motion (fBm) and standard Brownian motion (Bm). The integral with respect to Bm is the standard Ito integral, and the integral with respect to fBm is the generalised Riemann-Stieltjes integral using the tools of fractional calculus. An averaging principle in which the fast-varying diffusion process of the fast-slow systems acts as a noise to be averaged out in the limit is established. It is shown that the slow process has a limit in the mean square sense, which is characterized by the solution of stochastic differential equation…
$L^2$-cutoff for the averaging process on random regular graphs
研究了随机正则图上平均过程的$L^2$-截断，发现当度数d=10时出现相变，小度数下混合速度快，大度数下混合机制不同。
Strong averaging principle for multiscale time-inhomogeneous SDEs with multiplicative $α$-stable noises
本文针对乘性α-稳定噪声驱动的多尺度非齐次时间随机系统，基于Khasminskii离散化方法建立强平均原理，证明慢子系统向两类平均系统的强收敛性，并将结果应用于气候-天气系统。
We prove a stochastic averaging theorem for stochastic differential equations in which the slow and the fast variables interact. The approximate Markov fast motion is a family of Markov process with generator ${\mathcal L}_x$ for which we obtain a locally uniform law of large numbers and obtain the continuous dependence of their invariant measures on the parameter $x$. These results are obtained under the assumption that ${\mathcal L}_x$ satisfies Hörmander's bracket conditions, or more generally ${\mathcal L}_x$ is a family of Fredholm operators with sub-elliptic estimates. On the other hand …
Mathematical models of biological neural networks are associated to a rich and complex class of stochastic processes. In this paper, we consider a simple {\em plastic} neural network whose {\em connectivity/synaptic strength} $(W(t))$ depends on a set of activity-dependent processes to model {\em synaptic plasticity}, a well-studied mechanism from neuroscience. A general class of stochastic models has been introduced in \cite{robert_mathematical_2020} to study the stochastic process $(W(t))$. It has been observed experimentally that its dynamics occur on much slower timescale than that of the …
Averaging principle for slow-fast systems of rough differential equations via controlled paths
本文在受控路径框架下，证明了由独立随机粗糙路径驱动的慢快粗糙微分方程系统的强平均化原理，并利用Khas'minskii时间离散化方法完成证明。
In this work we study the averaging principle for non-autonomous slow-fast systems of stochastic differential equations. In particular in the first part we prove the averaging principle assuming the sublinearity, the Lipschitzianity and the Holder's continuity in time of the coefficients, an ergodic hypothesis and an $\mathcal{L}^2$-bound of the fast component. In this setting we prove the weak convergence of the slow component to the solution of the averaged equation. Moreover we provide a suitable dissipativity condition under which the ergodic hypothesis and the $\mathcal{L}^2$-bound of the…
Almost Sure Averaging for Fast-slow Stochastic Differential Equations via Controlled Rough Path
针对 Hurst 指数在1/3到1/2间的分数布朗运动驱动的快慢随机微分方程耦合系统，结合时间离散化、受控粗糙路径与随机动力系统方法，证明慢分量解几乎必然收敛到平均方程的解。
Fast transport asymptotics for stochastic RDEs with boundary noise
考虑一类在边界上也具有随机扰动的随机反应扩散方程，证明了当扩散速率远大于反应速率时，可以用一个合适的一维随机微分方程代替SPDE.这种替换在扩散的谱隙假设下是可能的，并且是快速空间传输中平均的结果。我们还研究了平均运动周围的波动。
Asymptotic-preserving approximations for stochastic incompressible viscous fluids and SPDEs on graph
针对含快对流项的多尺度随机反应-扩散-对流方程，提出一种渐近保持指数欧拉近似格式，通过三项关键理论推导证明其渐近保持性，并通过数值实验验证了理论结果。
Averaging principles for nonautonomous multiscale McKean-Vlasov stochastic systems
研究非自治多尺度McKean-Vlasov随机系统，利用非自治泊松方程建立强和弱平均原理并给出显式收敛率，在快尺度系数渐近收敛或时间周期条件下慢分量收敛到与ε无关的平均方程。
Averaging principles for time-inhomogeneous multi-scale SDEs via nonautonomous Poisson equations
本文研究时间非齐次多尺度随机微分方程的渐近行为，通过分析马尔可夫半群测度演化与非自治泊松方程性质，推导了两类带显式收敛速率的强弱平均原理，并通过算例验证了有效性。
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