Xue-Mei Li, Julian Sieber
We prove a fractional averaging principle for interacting slow-fast systems. The mode of convergence is in Hölder norm in probability. The main technical result is a quenched ergodic theorem on the conditioned fractional dynamics. We also establish geometric ergodicity for a class of fractional-driven stochastic differential equations, improving a recent result of Panloup and Richard.
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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。
Large deviation principle for slow-fast system with mixed fractional Brownian motion
本文针对 Hurst 参数大于 1/2 的混合分数布朗运动驱动的慢-快系统，结合变分框架、弱收敛方法与 Khasminskii 平均化原理，建立了慢分量的大偏差原理。
On the slow points of fractional Brownian motion
本文提出新方法计算分数布朗运动慢点的Hausdorff维数，基于SPDE慢增长点思想并引入新的局部化技术。
Large deviation for slow-fast McKean-Vlasov stochastic differential equations driven by fractional Brownian motions and Brownian motions
本文研究了由布朗运动和分数布朗运动驱动的慢-快McKean-Vlasov随机微分方程，提出乘积空间上的大偏差原理新定义，并在适当假设下利用弱收敛方法证明了该方程的LDP。
Large deviation principle for slow-fast systems with infinite-dimensional mixed fractional Brownian motion
本文研究了受无限维混合分数布朗运动扰动的慢-快系统的大偏差原理，采用基于变分表示公式的弱收敛方法，结合Khasminskii平均原理与时间离散化技术，去除了系数有界性假设，并进一步建立了中偏差原理。
Derivation of non-polynomial fractional diffusions from the generalized exclusion with a slow barrier
本文在流体动力学极限下，从长程相互作用且跳跃率依赖局部占据的随机粒子系统推导出广义分数阶多孔介质方程，并研究了慢势垒对质量流动的阻碍效应。
This article studies typical dynamics and fluctuations for a slow-fast dynamical system perturbed by a small fractional Brownian noise. Based on an ergodic theorem with explicit rates of convergence, which may be of independent interest, we characterize the asymptotic dynamics of the slow component to two orders (i.e., the typical dynamics and the fluctuations). The limiting distribution of the fluctuations turns out to depend upon the manner in which the small-noise parameter is taken to zero relative to the scale-separation parameter. We study also an extension of the original model in which…
We study the Hölderian regularity of Gaussian wavelets series and show that they display, almost surely, three types of points: slow, ordinary and rapid. In particular, this fact holds for the Fractional Brownian Motion. We also show that this property is satisfied for a multifractal extension of Gaussian wavelet series. Finally, we remark that the existence of slow points is specific to these functions.
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…
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.
This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: \[ \left(\partial^β+\fracν{2}(-Δ)^{α/2}\right)u(t,x) = I_t^γ\left[ρ(u(t,x))\dot{W}(t,x)\right],\quad t>0,\: x\in\mathbb{R}^d, \] where $\dot{W}$ is the space-time white noise, $α\in(0,2]$, $β\in(0,2)$, $γ\ge 0$ and $ν>0$. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang's condition: $d<2α+\fracαβ\min(2γ-1,0)$. In …
Large deviations of slow-fast systems driven by fractional Brownian motion
本文研究慢分量受分数布朗运动扰动、快分量由布朗运动驱动的多尺度随机系统，利用Young积分与分数阶微积分建立均匀化极限下的大偏差原理，并揭示速率函数在H=1/2处的不连续性。
We analyze the effect of additive fractional noise with Hurst parameter $H > \frac{1}{2}$ on fast-slow systems. Our strategy is based on sample paths estimates, similar to the approach by Berglund and Gentz in the Brownian motion case. Yet, the setting of fractional Brownian motion does not allow us to use the martingale methods from fast-slow systems with Brownian motion. We thoroughly investigate the case where the deterministic system permits a uniformly hyperbolic stable slow manifold. In this setting, we provide a neighborhood, tailored to the fast-slow structure of the system, that conta…
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…
Averaging Dynamics and Wong-Zakai approximations for a Fast-Slow Navier-Stokes System Driven by fractional Brownian Motion
本文研究了由分数布朗运动驱动的快慢纳维-斯托克斯系统，通过粗糙路径理论分析其极限行为，揭示了不同H值下慢分量收敛到不同类型的纳维-斯托克斯系统的特性。
We study the nonlinear stochastic time-fractional diffusion equations in the spatial domain $\mathbb{R}$, driven by multiplicative space-time white noise. The fractional index $β$ varies continuously from $0$ to $2$. The case $β=1$ (resp. $β=2$) corresponds to the stochastic heat (resp. wave) equation. The cases $β\in \:]0,1[\:$ and $β\in \:]1,2[\:$ are called {\it slow diffusion equations} and {\it fast diffusion equations}, respectively. Existence and uniqueness of random field solutions with measure-valued initial data, such as the Dirac delta measure, are established. Upper bounds on all $…
Central limit theorem for slow-fast system under mixed fractional Brownian motion
研究在分数布朗运动驱动的慢-快系统中，通过Young-Wiener意义下的解映射，分析平均极限的波动，涉及非马尔可夫性设定和Gronwall型结果。
We study statistical inference for small-noise-perturbed multiscale dynamical systems where the slow motion is driven by fractional Brownian motion. We develop statistical estimators for both the Hurst index as well as a vector of unknown parameters in the model based on a single time series of observations from the slow process only. We prove that these estimators are both consistent and asymptotically normal as the amplitude of the perturbation and the time-scale separation parameter go to zero. Numerical simulations illustrate the theoretical results.
In this paper, we use local fraction derivative to show the Hölder continuity of the solution to the following nonlinear time-fractional slow and fast diffusion equation: $$\left(\partial^β+\fracν{2}(-Δ)^{α/2}\right)u(t,x) = I_t^γ\left[σ\left(u(t,x)\right)\dot{W}(t,x)\right],\quad t>0,\: x\in\mathbb{R}^d,$$ where $\dot{W}$ is the space-time white noise, $α\in(0,2]$, $β\in(0,2)$, $γ\ge 0$ and $ν>0$, under the condition that $2(β+γ)-1-dβ/α>0$. The case when $β+γ\le 1$ has been obtained in \cite{ChHuNu19}. In this paper, we have removed this extra condition, which in particular includes all cases…
Moderate deviation principle for multiscale systems driven by fractional Brownian motion
本文研究了由分数布朗运动驱动的慢-快随机动力学系统的中偏差原理，推导了其成立的条件，并揭示了受fBm扰动的系统尾部行为与受标准布朗运动扰动的系统存在不同特征。
Homogenization of conditional slow-fast McKean-Vlasov SDEs
本文研究完全耦合的条件慢-快McKean-Vlasov随机微分方程，在不作周期性假设下，利用Wasserstein空间中的扰动方法推导其到同质化极限的收敛速率。
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.
Large deviation principle for slow-fast rough differential equations via controlled rough paths
本文在受控粗糙路径框架下，针对由混合分数布朗运动驱动的慢-快粗糙微分方程证明了大偏差原理，通过变分表示将问题转化为受控系统的定性性质，并利用快方程不变测度平均与解映射连续性证明受控慢分量的弱收敛。
The central limit theorem for slow-fast systems with Lévy noise
本文研究带 Lévy 噪声的慢-快随机微分系统，运用扰动测试函数方法分析其正态偏差，证明偏差可由高斯过程近似并建立中心极限定理。
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.
We review recent developments of slow/fast stochastic differential equations, and also present a new result on Diffusion Homogenisation Theory with fractional and non-strong-mixing noise and providing new examples. The emphasise of the review will be on the recently developed effective dynamic theory for two scale random systems with fractional noise: Stochastic Averaging and `Rough Diffusion Homogenisation Theory'. We also study the geometric models with perturbations to symmetries.
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.
The Large Deviation Principle for Stochastic Flow of Stochastic Slow-Fast Motions
本文研究完全耦合慢快运动中慢变量的随机流，证明其存在性并满足大偏差原理，方法基于Khasminskii平均原理、变分表示和弱收敛框架。
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 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.
Fluctuations from a random fractional averaging limit
本文研究了由分数布朗运动驱动的多重尺度随机微分方程系统，通过分数平均和同调定理，推导了慢运动波动收敛于由分数布朗运动和空间-时间高斯场驱动的随机微分方程的结论。
On the study of slow-fast dynamics, when the fast process has multiple invariant measures
本文研究快过程具有多个不变测度的慢-快随机系统平均化问题，在解耦与完全耦合及有限状态连续时间马尔可夫快过程设定下，证明极限动力学为随机 ODE。
Fractional stochastic volatility models have been widely used to capture the non-Markovian structure revealed from financial time series of realized volatility. On the other hand, empirical studies have identified scales in stock price volatility: both fast-time scale on the order of days and slow-scale on the order of months. So, it is natural to study the portfolio optimization problem under the effects of dependence behavior which we will model by fractional Brownian motions with Hurst index $H$, and in the fast or slow regimes characterized by small parameters $\eps$ or $δ$. For the slowly…
We design numerical schemes for a class of slow-fast systems of stochastic differential equations, where the fast component is an Ornstein-Uhlenbeck process and the slow component is driven by a fractional Brownian motion with Hurst index $H>1/2$. We establish the asymptotic preserving property of the proposed scheme: when the time-scale parameter goes to $0$, a limiting scheme which is consistent with the averaged equation is obtained. With this numerical analysis point of view, we thus illustrate the recently proved averaging result for the considered SDE systems and the main differences wit…
Almost Sure Averaging for Evolution Equations driven by fractional Brownian motions
本文研究由两类不同 Hurst 参数 FBM 分别驱动慢、快分量的耦合发展方程，建立逐轨道温和解，并借助随机不动点与时间离散化证明慢分量几乎必然收敛到平均化方程。
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。
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 local nondeterminism for stochastic time-fractional slow and fast diffusion equations
本文研究了随机时间分数慢扩散和快扩散方程的强局部非确定性，推导了存在随机场解的精确条件，并给出了时间与空间增量的方差界及非确定性结果。
Fast convergence rates for estimating the stationary density in SDEs driven by a fractional Brownian motion with semi-contractive drift
本文提出了一种基于核的估计器，用于估计由分数布朗运动驱动且具有半收缩漂移的SDE的平稳密度，通过鞅分解和新的收敛界限实现了更快的收敛速率，并展示了在离散观测下的应用及数据驱动的带宽选择方法。
Isotropic Q-fractional Brownian motion on the sphere: regularity and fast simulation
本文提出球面上各向同性 Q-分数布朗运动，证明其时空 Hölder 正则性，用谱方法近似并证明收敛，通过循环嵌入或条件化随机中点位移快速模拟，数值验证了时间精度与复杂度。
A large deviation principle is derived for stochastic partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.
This work is concerned with the dynamics of a slow-fast stochastic evolutionary system quantified with a scale parameter. An invariant foliation decomposes the state space into geometric regions of different dynamical regimes, and thus helps understand dynamics. A slow invariant foliation is established for this system. It is shown that the slow foliation converges to a critical foliation (i.e., the scale parameter is zero) in probability distribution, as the scale parameter tends to zero. The approximation of slow foliation is also constructed with error estimate in distribution. Furthermore,…
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