## 金融代写|随机偏微分方程代写Stochastic partialApproximative Centre Manifold

$$\partial_{t} u=\Delta u+\xi,$$

statistics-lab™ 为您的留学生涯保驾护航 在代写随机偏微分方程方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机偏微分方程代写方面经验极为丰富，各种代写随机偏微分方程相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Approximative Centre Manifold

This section describes how the evolution of solutions of a stochastic PDE subject to additive forcing is determined by an approximate centre manifold. This was briefly discussed in [Blö03] for the first order approximation. There the manifold is just

the vector space $\mathcal{N}$. It attracts solutions up to errors of order $\mathcal{O}\left(\varepsilon^{2}\right)$, and the flow along $\mathcal{N}$ is given on the slow time-scale by the amplitude equation.

Here, we first state the results of Section $4.1$ in [BH05], which relies on the second order correction introduced in [BH04] to describe the distance from $\mathcal{N}$, too. Therefore we need nonlinear stability, in order to bound moments. That is why we restrict ourselves in the following to nonlinear stable equations given by Assumption $2.7$.

The key difference from results on random invariant manifolds (cf. for example [DLS03] or [MZZ07; DLS04; DW06b]) is that we obtain in first order $\mathcal{O}(\varepsilon)$ a fixed object, instead of a random set that is moving in time. Our result allows to control this dynamics at least to order $\mathcal{O}\left(\varepsilon^{2}\right)$ or $\mathcal{O}\left(\varepsilon^{3}\right)$. We pay for that qualitative description, by having all statements just with high probability, and not almost surely.

Our main result shows that in first order the flow of solutions of the SPDE (1.2) along $\mathcal{N}$ is well approximated by $\varepsilon a\left(\varepsilon^{2} t\right)$, where $a$ is the solution of the amplitude equation. In second order $\mathcal{O}\left(\varepsilon^{2}\right)$, the distance from $\mathcal{N}$ is given by fast oscillations, which is given as a stationary Ornstein-Uhlenbeck process $\varepsilon^{2} \psi^{\star}(t)$. Thus the solutions are attracted by an $\mathcal{O}\left(\varepsilon^{3-\kappa}\right)$-neighbourhood of $\varepsilon^{2} \psi^{\star}(t)+\mathcal{N}$. Note that everything is valid only with high probability. Note that the setting for multiplicative noise is simpler, as the deterministic fixed point 0 is available. Therefore local results on the structure of invariant manifolds were obtained much earlier in that case (cf. for example [CLR01]). In Figure $3.2$ the typical behaviour of solutions is given.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Random Fixed Points

Let us discuss the dynamics of random fixed points for random dynamical systems induced by the SPDE. At this point we do not give a precise definition of random dynamical systems (see [Arn98] for details) or random fixed points (see for example [Sch98]). In this section it is enough to know that a random fixed point induces a stationary solution for the SPDE, if we start the SPDE in the random fixed point.
Theorem 3.7 Suppose Assumptions 2.5, 2.7, and $2.8$ are true, and let $u^{}(t)$ be a stationary mild solution in $X$ of (1.2). Let a be the solution of (1.5) with $a(0)=\varepsilon^{-1} P_{c} u^{}(0)$. Furthermore let $\psi^{\star}$ be the stationary Ornstein-Uhlenbeck given by (3.25).

Then there is a constant $c_{0}>0$ such that for all $T_{0}>0$ any small $\kappa \in(0,1)$, and all $p \geq 1$, there exists a constant $C>0$, such that
$$\mathbb{E}\left(\sup {t \in\left[c{0} \ln (1 / \varepsilon), T_{0} / \varepsilon^{2}\right]}\left|u^{}(t)-\varepsilon a\left(t \varepsilon^{2}\right)-\varepsilon^{2} \psi^{\star}(t)\right|_{X}^{p}\right)^{1 / p} \leq C \varepsilon^{3-\kappa} .$$ This result is an extension of the result for invariant measures (cf. Theorem 3.1), as the law of the stationary solution is at a fixed time $t$ exactly an invariant measure. Here we can control the time-evolution, too. Idea of proof: First we start the approximation result with initial condition $u^{}\left(-T_{a} \varepsilon^{-2}\right)$ for some $T_{a}>0$. This implies first for $t=0$, but then due to stationarity for all $t$, that
$$\left(\mathbb{E}\left|u^{}(t)\right|^{p}\right)^{1 / p} \leq C \varepsilon \quad \text { and } \quad\left(\mathbb{E}\left|P_{s} u^{}(t)\right|^{p}\right)^{1 / p} \leq C \varepsilon^{2} .$$
Thus, we can start the approximation result in 0 . and get
$$\mathbb{E}\left(\sup {t \in\left[0, T{0} / \varepsilon^{2}\right]}\left|u^{}(t)-\varepsilon a\left(t \varepsilon^{2}\right)-\varepsilon^{2} \psi(t)\right|_{X}^{p}\right)^{1 / p} \leq C \varepsilon^{3-\kappa},$$ where $\psi$ is the OU-process with initial condition $\psi(0)=\varepsilon^{-2} P_{s} u^{}(0)$.
After a time scale of oder $\mathcal{O}(\ln (1 / \varepsilon))$ we can approximate $\psi$ with $\psi^{}$ as $$\psi(t)=\mathrm{e}^{t L}\left(\psi(0)-\psi^{}(0)\right)+\psi^{*}(t)$$

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Random Set Attractors

Let us extend our result for random fixed points (cf. Section 3.3.1) to random attractors. We do not present the result in full detail, but rather focus on a brief description of all steps necessary. Most steps are just quite technical but straightforward extensions of the estimates necessary for the residual, attractivity, and approximation. The key point is to rely on path-wise estimates, and take expectations in the end.
Assumption 3.3 Consider eq. (1.2) fulfilling Assumptions 2.5, 2.7, and 2.8.
The main example, we keep in mind is the stochastic Swift-Hohenberg equation in the space $X=L^{2}(G)$.

First we can use standard a priori estimates relying on nonlinear stability. This is very similar to Theorem $2.8$, but nevertheless, we need to get uniform bounds with respect to the initial conditions
$$u(0) \in B_{r}:={x \in X:|x| \leq r}$$
for any fixed $r>0$. For this we establish path-wise bounds for $v=u-\varepsilon^{2} \phi$ with $\phi=W_{L-\varepsilon^{2}}$, which solves the following random PDE (compare (2.87))
$$\partial_{t} v=L v+\varepsilon^{2}(A v+\phi)+\varepsilon^{4} A \phi+\mathcal{F}\left(v+\varepsilon^{2} \phi\right), \quad v(0)=u(0) .$$
We use standard deterministic a priori estimates for (3.26), and take expectations in the end. Note that this transformation is not ergodic, in contrast to the usual transformation in the theory of random attractors, where one uses the stationary Ornstein-Uhlenbeck process for $\phi$. For our setting we rely on $\phi$, as we do not want to change initial conditions in (3.26).

The first step is the attractivity. It is similar to the proof of Theorem $2.8$ and follows from standard a priori estimates for $v$ and the stochastic convolution. Note that we rely on nonlinear stability. The result is that for all $r>0$ there is a time $T_{e}=\mathcal{O}\left(\varepsilon^{-2}\right)$ such that for all $p>0$
$$\mathbb{E}\left(\sup {u(0) \in B{r}}|u(t)|^{p}\right) \leq C \varepsilon^{p} \text { for all } t \geq T_{e} .$$

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Random Fixed Points

(和|在(吨)|p)1/p≤Ce 和 (和|磷s在(吨)|p)1/p≤Ce2.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Random Set Attractors

∂吨在=大号在+e2(一种在+φ)+e4一种φ+F(在+e2φ),在(0)=在(0).

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|随机偏微分方程代写Stochastic partialApplications Some Examples

$$\partial_{t} u=\Delta u+\xi,$$

statistics-lab™ 为您的留学生涯保驾护航 在代写随机偏微分方程方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机偏微分方程代写方面经验极为丰富，各种代写随机偏微分方程相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Some Examples

This chapter presents applications of the approximation via amplitude equations. The main results are about long-time behaviour of SPDEs or transient pattern formation for SPDEs on bounded domains. For simplicity of presentation, we focus on a few examples, in order to highlight the key ideas.

By no means we give an exhaustive presentation of all results possible, but focus on three examples. First we treat approximation of invariant measures near a change of stability. This is a review of results of [BH04]. We give the main ideas without stating details of the proofs.

The second section on pattern formation below threshold of instability gives a self-contained introduction, by explaining ideas and giving all proofs in the simplest setting possible. The final section on approximative centre manifolds and approximation of random attractors gives only the main ideas of proofs.
Invariant Measures
Section $3.1$ states the approximation of invariant measures for the corresponding dual Markov semigroup. We summarise some of the results of [BH04]. Near the change of stability the invariant measure is well described in first order of $\varepsilon$ by the invariant measure of the amplitude equation plus in second order by an infinite dimensional Ornstein-Uhlenbeck measure on the stable modes $\mathcal{S}$. The result is of the type $\mathbb{P}^{u^{}}=\mathbb{P}^{e a^{}} \otimes \mathbb{P}^{e^{2} \psi^{*}}$. In this part the presentation is based on the setting and the results of Section 2.5. Apart from large deviation results, this is the first rigorous qualitative result for the structure of invariant measures for SPDEs with additive noise.

Another interesting application, that we nevertheless do not treat here, is the discussion of phenomenological bifurcation for SPDEs. It relies on the approximation of invariant measures. The invariant measure in $\mathcal{N}$ for the amplitude equation is usually easy to describe. For instance one can use the celebrated Fokker-Planck equation (cf. Risken [Ris84]), where we identify $\mathcal{N}$ with some $\mathbb{R}^{n}$ with $n \in{1,2}$ for many examples. The Fokker-Planck equation is a deterministic PDE, which solution provides a smooth Lebesgue density of invariant measures on $\mathcal{N}$.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Approximation of Invariant Measures

This section reviews the results obtained in [BH04] on approximating the invariant measure of SPDEs of the type of (1.2) near a change of stability. For simplicity the result is based on the setting of Section 2.5, where we considered a stable cubic nonlinearity and additive noise. To be more precise, consider equation (1.2) and let Assumptions 2.5, 2.7, and $2.8$ be true. Additive noise is important, in order to have a unique exponential attracting invariant measure for the amplitude equation (cf. Assumption $3.1$ and the discussion below).

It is a main issue to have the speed of convergence to the invariant measure for the amplitude equations under control (see (3.7)). The flow has to be (up to small errors) a contraction on the space of probability measures. This makes multiplicative noise more complicated, as there could be more than one invariant measure, and the speed of convergence is not controlled, as even nearby initial conditions may converge to different measures. A similar problem arises, when the amplitude equation is deterministic, for example, if the noise strength in the SPDE is $\mathcal{O}\left(\varepsilon^{3}\right)$. Here only partial results are available. Again problems arise with the speed of convergence in the amplitude equation, once its deterministic attractor is not trivial.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|The Results

Before we state our results, we introduce one more notation. For simplicity of presentation, we rescale the solutions of (1.2) by $\varepsilon$ such that they are concentrated on a set of order 1 instead of a set of order $\varepsilon$. Furthermore, we rescale the equation to the slow time-scale $T=t \varepsilon^{2}$. Thus we consider $v$ given by $v(T)=\varepsilon^{-1} u\left(T \varepsilon^{-2}\right)$, where we split as usual $v=v_{c}+v_{s}\left(v_{c} \in \mathcal{N}, v_{s} \in \mathcal{S}\right)$. We obtain
$$\begin{array}{lc} \partial_{T} v_{c}= & A_{c}\left(v_{c}+v_{s}\right)+\mathcal{F}{c}\left(v{c}+v_{s}\right)+\partial_{T} \beta \ \partial_{T} v_{s}=\varepsilon^{-2} L v_{s}+A_{s}\left(v_{c}+v_{s}\right)+\mathcal{F}{s}\left(v{c}+v_{s}\right)+\partial_{T} \hat{W}{s} \end{array}$$ where $\hat{W}{s}(T)=\varepsilon P_{s} Q W\left(\varepsilon^{-2} T\right)$ and $\beta(T)=\varepsilon P_{c} Q W\left(\varepsilon^{-2} T\right)$, as usual.
We denote by $\mu_{\star}^{e}$ an invariant measure of (3.4). Note that the existence is standard using the celebrated Krylov-Bogoliubov method (cf. [DPZ96]).

Definition 3.5 Denote by $\nu_{\star}^{}$ the invariant measure for the pair of processes $(a, \varepsilon \psi)$, where the evolution is given by $(1.5)$ and (1.6). Hence, in the slow time variable \begin{aligned} &\partial_{T} a=A_{c} a+\mathcal{F}{c}(a)+\partial{T} \beta \ &\partial_{T} \psi=\varepsilon^{-2} L \psi+\partial_{T} \hat{W}{s} \end{aligned} Denote by $\nu{\star}^{c}$ the marginal on $\mathcal{N}$, and by $\nu_{\star}^{s}$ the one on $\mathcal{S}$, respectively.
Note that we actually do not need the uniqueness of $\nu_{\star}^{}$. We only need that the marginals on $\mathcal{N}$ and $\mathcal{S}$ are unique. The uniqueness of $\nu_{\star}^{s}$ is obvious, as we have an Ornstein-Uhlenbeck process. Furthermore, the uniqueness of $\nu_{\star}^{c}$ follows from Assumption 3.1.

Note that $\nu_{\star}^{*}$ depends on $\varepsilon$ by the rescaling of $\psi$. Recall also that we discussed in Remark $2.8$ that the two noise terms in (3.5) may not be independent. Thus the equations in (3.5) are coupled through the noise, but actually they do not live on the same time scale, as the second equation in (3.5) lives on the fast time-scale $t$. However, as the equations are otherwise decoupled, we can determine the marginals $\nu_{\star}^{c}$ and $\nu_{\star}^{s}$ independently. The marginal $\nu_{\star}^{\mathrm{c}}$ is independent of $\varepsilon$ and $\nu_{\star}^{s}$ depends on $\varepsilon$ only through the trivial scaling of $\varepsilon \psi$. Therefore we suppressed this $\varepsilon$-dependence in the notation.

With these notations, our main result in the Wasserstein distance is the following.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|The Results

$$\begin{array}{lc} \partial_{T} v_{c}= & A_{c}\left(v_{c}+v_{s}\right)+\mathcal{F} {c }\left(v {c}+v_{s}\right)+\partial_{T} \beta \ \partial_{T} v_{s}=\varepsilon^{-2} L v_{s}+A_{ s}\left(v_{c}+v_{s}\right)+\mathcal{F} {s}\left(v {c}+v_{s}\right)+\partial_{T} \hat{ W} {s} \end{array}$$ 其中 $\hat{W} {s}(T)=\varepsilon P_{s} QW\left(\varepsilon^{-2} T\right)一种nd\beta(T)=\varepsilon P_{c} QW\left(\varepsilon^{-2} T\right),一种s在s在一种l.在和d和n这吨和b是\mu_{\star}^{e}$ (3.4) 的不变测度。请注意，存在是使用著名的 Krylov-Bogoliubov 方法（参见 [DPZ96]）的标准。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|随机偏微分方程代写Stochastic partialAssumptions

$$\partial_{t} u=\Delta u+\xi,$$

statistics-lab™ 为您的留学生涯保驾护航 在代写随机偏微分方程方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机偏微分方程代写方面经验极为丰富，各种代写随机偏微分方程相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Assumptions

Let us summarise all assumptions necessary for our results. We do not focus on the highest possible level of generality, but stick to some simpler setting which cover all our examples. First consider the linear operator $L$.

Assumption 2.5 Let $X$ be a separable Hilbert space and $\Delta$ (subject to some boundary conditions on a bounded domain) be a self-adjoint version of the Laplacian on $X$. Suppose $L=P(-\Delta)$ for some function $P$ such that $L$ is non-positive. Furthermore, let the kernel $\mathcal{N}=\operatorname{ker} L$ of $L$ be non-empty and finite dimensional. Finally, suppose $P(k) \rightarrow-\infty$ as $k \rightarrow \infty$.

This assumption is a stronger than the one in Section 2.2. It is mainly used for convenience of presentation, and covers all examples presented. Furthermore, it is just a special case of Assumption 2.1, and in the following we can use all the implications of this assumption. We use the notation $P_{c}$ and $P_{s}$, which are in this case just the standard orthogonal projections. Additionally, recall the splitting $X=\mathcal{N} \oplus \mathcal{S}$ with $\mathcal{S}=P_{s} X$ and the spaces $X^{\alpha}$ from Section 2.2. Recall furthermore the bounds (2.4), (2.5), and (2.6) for the analytic semigroup $\mathrm{e}^{t L}$ generated by $L$.
For the nonlinearities, we make two assumptions. The first one, is much weaker than Assumption 2.2, as we are aiming only for local results in that case. Especially, we can get rid of the strong nonlinear dissipativity. The second assumption is similar to Assumption $2.2$ and involves strong nonlinear stability and dissipativity conditions in $\mathcal{N}$.

Assumption 2.6 The function $\mathcal{F}$ is locally Lipschitz from $X$ to $X^{-\alpha}$ for some $\alpha \in[0,1)$. This means that for all $R>0$ there is a $C>0$ such that
$$\left|\mathcal{F}\left(v_{1}\right)-\mathcal{F}\left(v_{2}\right)\right|_{-\alpha} \leq C\left|v_{1}-v_{2}\right| \quad \text { for all } v_{i} \text { with }\left|v_{i}\right| \leq R$$
Assume we can split $\mathcal{F}(x)=f(x)+g(x)$, where $f: X \times X \times X \rightarrow X^{-\alpha}$ is continuous, trilinear, and symmetric. The function $g$ is of higher order, which means $|g(x)|_{-\alpha} \leq C|x|^{4}$ provided $|x| \leq 1 .$

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Amplitude Equations Main Results

Theorem 2.7 (Attractivity-local) Under Assumptions $2.5,2.6$, and $2.8$ fix some small constant $\kappa>0$. Then there are constants $c_{i}>0$ and a time $t_{e}=$ $\mathcal{O}\left(\ln \left(\varepsilon^{-1}\right)\right)$ such that for all mild solutions $u$ of $(2.86)$ we can write
$$u\left(t_{e}\right)=\varepsilon a_{e}+\varepsilon^{2} R_{e} \quad \text { with } \quad a_{\varepsilon} \in \mathcal{N} \text { and } R_{e} \in \mathcal{S}$$
where
$$\mathbb{P}\left(\left|a_{e}\right| \leq \delta,|R e| \leq \varepsilon^{-\kappa}\right) \geq \mathbb{P}\left(|u(0)| \leq c_{3} \delta \varepsilon\right)-c_{1} e^{-c_{2} e^{-2 \kappa}}$$
for all $\delta>1$ and $\varepsilon \in(0,1)$.
This result states in a weak form that $u(0)=\mathcal{O}(\varepsilon)$ with high probability implies $P_{c} u\left(t_{\varepsilon}\right)=\mathcal{O}(\varepsilon)$ and $P_{s} u\left(t_{\varepsilon}\right)=\mathcal{O}\left(\varepsilon^{2}\right)$ with high probability, too. Note that we do not bound any moments of the solution $u$.

We do not give a detailed proof of this result, as it is a straightforward modification of Theorem $3.3$ of [Blö03]. It relies on the fact that small solutions of order $\mathcal{O}(\varepsilon)$ are on small time-scales given by the linearised picture, which is dominated by the semigroup estimates $(2.5)$ and $(2.6)$. Thus modes in $P_{s} X$ decay exponentially fast on a time-scale of order $\mathcal{O}(1)$.
Using strong nonlinear stability, we can prove much more:
Theorem 2.8 (Attractivity-global) Let Assumptions 2.5, 2.6, and 2.8 be satisfied. Then for all times $T_{e}=T_{0} \varepsilon^{-2}>0$ and for all $p \geq 1$ there are constants $C_{p}>0$ explicitly depending on $p$ such that
$$\mathbb{E}\left|u\left(t+T_{e}\right)\right|^{p} \leq C_{p} \varepsilon^{p} \quad \text { and } \quad \mathbb{E}\left|P_{s} u\left(t+T_{e}\right)\right|^{p} \leq C_{p} \varepsilon^{2 p}$$
for all $t \geq 0$, all $X$-valued mild solutions $u$ of equation (1.2) independent of the initial condition $u(0)$, and for all $\varepsilon \in(0,1)$.

Furthermore, if we already assume that $\mathbb{E}|u(0)|{ }^{p} \leq C_{p} \varepsilon^{p}$ for a constant $C_{p}>0$, then there is a time $t_{\varepsilon}=\mathcal{O}\left(\ln \left(\varepsilon^{-1}\right)\right)$ and a constant $C>0$ such that
$$\mathbb{E}|u(t)|^{p} \leq C \varepsilon^{p} \quad \text { and } \quad \mathbb{E}\left|P_{s} u\left(t+t_{e}\right)\right|^{p} \leq C \varepsilon^{2 p}$$
for all $t \geq 0$, all $X$-valued mild solutions $u$, and for all $\varepsilon \in(0,1)$.
The proof is given by a priori estimates. This was not directly proved in [BH04], but under our somewhat stronger assumptions this is similar to Lemma $4.3$ of [BH04]. It relies on a priori estimates for $v_{\delta_{\varepsilon}}=u-\varepsilon^{2} W_{L-\delta_{c}}$ with $\delta_{e}=\mathcal{O}\left(\varepsilon^{2}\right)$, which fulfils a random PDE similar to $(2.87)$. The main technical advantage is that the linear semigroup generated by $L-\delta_{e}$ is exponentially stable.
2.5.3.2 Approximation
For a solution $a$ of $(1.5)$ and $\psi$ of $(1.6)$ we define the approximations $\varepsilon w_{k}$ of order $k$ by
$$\varepsilon w_{1}(t):=\varepsilon a\left(\varepsilon^{2} t\right) \text { and } \varepsilon w_{2}(t):=\varepsilon a\left(\varepsilon^{2} t\right)+\varepsilon^{2} \psi(t)$$

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Quadratic Nonlinearities

In this section we review the results of [Blö05a]. Consider an SPDE of the following type.
$$\partial_{t} u=L u+\varepsilon^{2} A u+B(u, u)+\varepsilon^{2} \xi$$
where $L$ as in Assumption 2.1. The linear operator $A$ and the bilinear operator $B$ are as in Assumption 2.4, and the noise is the generalised derivative of some Wiener process (cf. Assumption 2.9).

In [Blö05a] we used fractional noise. This was motivated by the fact that the proofs rely on fractional integration by parts formulas, and explicit path-wise estimates. Here we state for simplicity only the version for Gaussian noise that is white in time. Note that due to the method of proof, we need the noise to be trace-class, as we need bounds for the Wiener process $W(t)$ in the space $X$. This obviously rules out space-time white noise.

Let us furthermore point out that the Hilbert space setting is not necessary in this approach, as we purely rely on local results, using cut-off techniques, and we do not use a priori estimates. It is also necessary to deal with non self-adjoint operators, as the linear part in the Rayleigh-Bénard system is not self-adjoint (cf. Section $6.1$ of [Blö05a] for a detailed discussion.
For the stochastic perturbation $\xi$ let the following assumption be true.
Assumption 2.9 (Noise) Suppose that the noise process $\xi$ is the generalised derivative of some Wiener process ${Q W(t)}_{t \geq 0}$ on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$, where $W$ is the standard cylindrical Wiener process.
Assume that the stochastic convolution
$$W_{L}(t)=\int_{0}^{t} \mathrm{e}^{(t-\tau) L} d Q W(\tau)$$
is a well defined stochastic process with continuous paths in $X$. We suppose that the noise (or $W)$ is of trace-class, i.e. $\operatorname{tr}\left(Q^{2}\right)=\mathbb{E}|Q W(t)|^{2}<\infty$.

This assumption is stronger than Assumption 2.8. Especially, $W$ being traceclass is a serious restriction, as this already implies that $W$ has continuous paths in $X$. We briefly sketched after Remark $2.7$ the connection between the spatial correlation function $q$ of the noise $\xi$ and the operator $Q$ belonging to $W$. The condition of $W$ being trace-class is essentially a regularity condition on $q$. See for example [Blö05b]. Any decay condition for the eigenvalues of $Q$ immediately transfers to a decay condition of the Fourier coefficients of $q$.
To give a meaning to $(2.93)$ we consider as usual mild solutions.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Assumptions

|F(在1)−F(在2)|−一种≤C|在1−在2| 对全部 在一世 和 |在一世|≤R

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Amplitude Equations Main Results

2.5.3.2 近似

e在1(吨):=e一种(e2吨) 和 e在2(吨):=e一种(e2吨)+e2ψ(吨)

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Quadratic Nonlinearities

∂吨在=大号在+e2一种在+乙(在,在)+e2X

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|随机偏微分方程代写Stochastic partialApproximation

$$\partial_{t} u=\Delta u+\xi,$$

statistics-lab™ 为您的留学生涯保驾护航 在代写随机偏微分方程方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机偏微分方程代写方面经验极为丰富，各种代写随机偏微分方程相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Approximation

For a solution $u$ of (2.51) define
$$a(0)=\varepsilon^{-1} P_{c} u(0) \quad \text { and } \quad \psi(0)=\varepsilon^{-2} P_{s} u(0)$$
Now let $a$ be a solution of (2.45) with initial condition $a(0)$ and define $w$ and $\psi$ as in $(2.47)$ and $(2.46)$. Then we can show that $u(t) \approx \varepsilon w(t)$ in the following sense:
Theorem 2.6 (Approximation) Let Assumptions 2.1, 2.3, and 2.4 be true.
For $\delta_{1}, \delta_{2}>0, \tilde{\kappa} \in(0,1]$, and $T_{0}>0$ there is some $\eta>0$ and some constant $C>0$ such that for all solutions $u$ of $(2.51)$ and approximations a and $\psi$ defined by (2.59), (2.47), and (2.46) we have
\begin{aligned} &\mathbb{P}\left(\sup {t \in\left[0, T{0} e^{-2}\right]}|u(t)-\varepsilon w(t)| \leq C \varepsilon^{2-\tilde{\kappa}}\right) \ &\geq 1-2 \mathbb{P}\left(\left|P_{s} u(0)\right|>\delta_{2} \varepsilon^{2}\right)-2 \mathbb{P}\left(\left|P_{c} u(0)\right|>\delta_{1} \varepsilon\right)-C \varepsilon^{\eta} \end{aligned}
for all $\varepsilon \in(0,1)$.
The proof will be given in the next section. Let us first comment on the improvements of the result compared to older results.

Remark 2.5 In the proof we need $|a(t)|^{2}$ to be bounded uniformly in $t \in\left[0, T_{0}\right]$ by $\gamma \ln \left(\varepsilon^{-1}\right)$ for some small $\gamma>0(c f$. $(2.76))$. Hence, as we rely on Theorem B. 9 we cannot improve the result to large $\eta>0$ (cf. equation (2.79)). The main obstacle is that we can only bound certain exponential moments of $|a|^{2}$ and not of higher powers. Nevertheless, this is an improvement to the results of [Blö05a], where the probability was just small without any order in $\varepsilon$. In principle it is easy to thoroughly compute all constants, in order to provide a uniform lower bound on $\eta$ independent of other constants like $\delta_{j}, T_{0}$, and $\kappa$. But, as we expect the bound not to be large, for simplicity of presentation we do not focus on that.

Remark 2.6 For special types of nonlinearities we will see from the proof that it is possible to improve the result of Theorem $2.6$ significantly. We need information about the sign of certain multi-linear functionals. The first one is of the type $F_{1}\left(a, a, R_{c}, R_{c}\right)=\left\langle B_{c}\left(\psi, R_{c}\right), R_{c}\right\rangle$, while the second is given by $F_{2}\left(a, a, R_{c}, R_{c}\right)=$ $\left\langle B_{c}\left(a, R_{s}\right), R_{c}\right\rangle+”$ Error”. Recall that $\psi$ depends quadratically on a, and we will see later in the proof that $R_{s}$ is a function of $R_{c}$. Thus a statement of these results is quite technical, but sometimes easy to check for given $B$ and $L$. The improvement is that we can use standard a priori type estimates for (2.70) and (2.71), where all the critical terms responsible for the bad order in the proof of Theorem $2.6$ disappear.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Proofs

This section gives the postponed proofs for Theorem 2.4, Lemma $2.2$, and Theorem 2.6. We first provide the proof of the attractivity result.

Proof. (of Theorem 2.4) The main ingredients of the proof are a cut-off technique and the linear stability of $L$. Fix some $\rho \in C^{\infty}(\mathbb{R})$ such that $\rho(x)=1$ for $x \leq 1$ and $\rho(x)=0$ for $x \geq 2$. Define for some small $\kappa>0$
$$A^{(\rho)}(u)=\rho\left(|u| \varepsilon^{-1+\kappa}\right) \cdot A u \quad \text { and } \quad B^{(\rho)}(u)=\rho\left(|u| \varepsilon^{-1+\kappa}\right) \cdot B(u) \text {. }$$
Moreover, define
$$u^{(\rho)}(0)=\left{\begin{array}{c} u(0): \text { for }|u(0)| \leq \delta \varepsilon \ 0: \text { otherwise } \end{array}\right.$$
Let $u^{(\rho)}$ be the solution of $(2.51)$ with $A^{(\rho)}$ and $B^{(\rho)}$ instead of $A$ and $B$ and initial condition $u^{(\rho)}(0)$. Thus,
\begin{aligned} u^{(\rho)}(t)=& \mathrm{e}^{t L} u^{(\rho)}(0)+\varepsilon \int_{0}^{t} \mathrm{e}^{(t-\tau) L} u^{(\rho)}(\tau) d \beta(\tau) \ &+\int_{0}^{t} \mathrm{e}^{(t-\tau) L}\left\varepsilon^{2} A^{(\rho)}\left(u^{(\rho)}\right)+B^{(\rho)}\left(u^{(\rho)}\right)\right d \tau . \end{aligned}
The existence of a unique solution $u^{(\rho)}$ is standard (cf. [DPZ92]), as we have global Lipschitz nonlinearities.
Define furthermore the stopping time
$$\tau_{\rho}=\left{\begin{array}{cc} \inf \left{t>0:\left|u^{(\rho)}(t)\right|>\varepsilon^{1-\kappa}\right} & : \text { for }|u(0)| \leq \delta \varepsilon \ 0 & : \text { otherwise. } \end{array}\right.$$
Obviously, $u(t)=u^{(\rho)}(t)$ for $0 \leq t \leq \tau_{\rho}$.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Setting for Additive Noise

In this section, we follow partly the presentation in [BH05], which reviews results of [Blö03] and [BH04], which in turn are based on [BMPSO1]. The setting is exactly the one sketched in Section 1.2. We focus on an SPDE of the type (1.2) with mild

solutions given by (1.19). The additive noise $\varepsilon^{2} \xi$ in the equation is for instance motivated by the presence of thermal fluctuations in the medium. Therefore the strength $\varepsilon^{2}$ of the noise is supposed to be very small. We usually assume that the noise $\xi=\partial_{t} W$ is some generalised Gaussian process, which is given by the derivative of some $Q$-Wiener process. We comment on that in detail later after Assumption $2.8$.

In Section 2.5.1 we summarise the precise mathematical assumptions for (1.2). The main results for the approximation via amplitude equations are given in Section 2.5.3. This setting is also used in Section $3.1$ to approximate long-time behaviour in terms of invariant measures.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Proofs

u^{(\rho)}(0)=\left{在(0): 为了 |在(0)|≤de 0: 除此以外 \对。 大号和吨在(ρ)b和吨H和s这l在吨一世这n这F(2.51)在一世吨H一种(ρ)一种nd乙(ρ)一世ns吨和一种d这F一种一种nd乙一种nd一世n一世吨一世一种lC这nd一世吨一世这n在(ρ)(0).吨H在s, \begin{aligned} u^{(\rho)}(t)=& \mathrm{e}^{t L} u^{(\rho)}(0)+\varepsilon \int_{0}^{t } \mathrm{e}^{(t-\tau) L} u^{(\rho)}(\tau) d \beta(\tau) \ &+\int_{0}^{t} \mathrm{ e}^{(t-\tau) L}\left \varepsilon^{2} A^{(\rho)}\left(u^{(\rho)}\right)+B^{(\rho) }\left(u^{(\rho)}\right)\right d \tau 。 \end{对齐} 吨H和和X一世s吨和nC和这F一种在n一世q在和s这l在吨一世这n在(ρ)一世ss吨一种nd一种rd(CF.[D磷从92]),一种s在和H一种在和Gl这b一种l大号一世psCH一世吨和n这nl一世n和一种r一世吨一世和s.D和F一世n和F在r吨H和r米这r和吨H和s吨这pp一世nG吨一世米和 \tau_{\rho}=\左{\begin{array}{cc} \inf \left{t>0:\left|u^{(\rho)}(t)\right|>\varepsilon^{1-\kappa}\right} & : \文本 { }|u(0)| \leq \delta \varepsilon \ 0 & : \text { 否则。} \结束{数组}\begin{array}{cc} \inf \left{t>0:\left|u^{(\rho)}(t)\right|>\varepsilon^{1-\kappa}\right} & : \文本 { }|u(0)| \leq \delta \varepsilon \ 0 & : \text { 否则。} \结束{数组}\对。

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Setting for Additive Noise

(1.19) 给出的解。附加噪声e2X例如，在等式中是由介质中存在的热波动引起的。因此实力e2的噪音应该很小。我们通常假设噪声X=∂吨在是一些广义的高斯过程，由一些的导数给出问-维纳过程。我们稍后会在假设之后对此进行详细评论2.8.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|随机偏微分方程代写Stochastic partialResults for Quadratic Nonlinearities

$$\partial_{t} u=\Delta u+\xi,$$

statistics-lab™ 为您的留学生涯保驾护航 在代写随机偏微分方程方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机偏微分方程代写方面经验极为丰富，各种代写随机偏微分方程相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Results for Quadratic Nonlinearities

This section states rigorous results for the approximation via amplitude equations for quadratic nonlinearities. We focus only on the interesting case, where $P_{c} B(a, a)=0$, which was discussed for additive noise on a formal level in Section 1.1.3. The case with $P_{c} B(a, a) \neq 0$ is similar to the cubic case. The formal result for our case is completely analogous to the one stated in Section 1.1.2, we summarise details below. Nevertheless, in this case in general we cannot bound moments of solutions. We have to use cut-off techniques in order to use moments.

Here we present a somewhat simpler model with multiplicative noise, in order to simplify the presentation. We review the results of [Blö05a] for additive noise in

Section 2.6. In [Blö05a] also fractional (i.e. smoother) additive noise was used, but we do not focus on that.
Consider
$$\partial_{t} u=L u+\varepsilon^{2} A u+B(u, u)+\varepsilon u \dot{\beta},$$
with $L$ and $A$ as in Assumption $2.1$ and $2.2$, and $B$ some bilinear mapping defined later on in Assumption 2.4.

Let us recall the formal derivation of the amplitude equation, which is similar to Section 1.1.3. Plugging the ansatz
$$u(t)=\varepsilon a\left(\varepsilon^{2} t\right)+\varepsilon^{2} \psi_{o}\left(\varepsilon^{2} t\right)$$
with $a \in \mathcal{N}$ and $\psi_{o} \in P_{s} X$ into (2.40), we derive in lowest order of $\varepsilon>0$
$\mathcal{O}\left(\varepsilon^{2}\right)$ in $\mathcal{N}: \quad 0=B_{c}(a, a)$,
$\mathcal{O}\left(\varepsilon^{3}\right)$ in $\mathcal{N}: \quad \partial_{T} a=A_{c} a+2 B_{c}\left(a, \psi_{o}\right)+a \partial_{T} \tilde{\beta}$,
$\mathcal{O}\left(\varepsilon^{2}\right)$ in $P_{s} X: \quad 0=L \psi_{o}+B_{s}(a, a) .$
Note that $\tilde{\beta}(T)=\varepsilon \beta\left(T \varepsilon^{-2}\right)$ is again a rescaled Brownian motion. From (2.41) we see that $B_{c}(a, a)=0\left(B_{c}:=P_{c} B\right.$, as usual $)$ is necessary for the approach presented. Finally, projecting (2.43) to $P_{s} X$ and solving for $\psi_{o}$ yields
$$\partial_{T} a=A_{c} a-2 B_{c}\left(a, L_{s}^{-1} B_{s}(a, a)\right)+a \partial_{T} \tilde{\beta}$$
or in integrated form
$$a(T)=a(0)+\int_{0}^{T}\leftA_{c} a-2 B_{c}\left(a, L_{s}^{-1} B_{s}(a, a)\right)\right d \tau+\int_{0}^{T} a(\tau) d \tilde{\beta}(\tau)$$
where we consider as before Itô-differentials. Nevertheless, as discussed before in Section 2.1, we could also consider Stratonovič-differentials everywhere, and still obtain the same result. An interesting feature of (2.45) is that the amplitude equation involves a cubic nonlinearity. Therefore, we can expect nonlinear stability of the amplitude equation, which is in general not present for the SPDE.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Attractivity

We use a cut-off technique, as in general we cannot control moments of solutions. There are some special cases like for instance one-dimensional Burgers, surface growth, or Kuramoto-Sivashinsky equation (see [BGR02; DPDT94; DE01]), where we actually can derive bounds for moments. But for our results it is enough to cut off the nonlinearity for large solutions, in order to keep it small for solutions that get too large.

This technique is well known for SDEs with blow-ups. See for example [McK69]. For a detailed discussion see Section $6.3$ of [HT94]. The idea is always to cut off the nonlinearities, in order to derive bounds for moments and to compute probabilities. But solutions of the modified equation with cut-off and the original equation coincide, as long as both are small. Note that for the local attractivity result we are anyway only interested in solutions that are small. To be more precise we look at solutions of order $\mathcal{O}(\varepsilon)$.

The main result is a local attractivity result for solutions of order $\mathcal{O}(\varepsilon)$. It shows that if $u(0)$ is of order $\mathcal{O}(\varepsilon)$, then at some time $t_{\varepsilon}=\mathcal{O}\left(\ln \left(\varepsilon^{-1}\right)\right)$ the probability is almost 1 that $u\left(t_{\varepsilon}\right)$ is still of order $\mathcal{O}(\varepsilon)$, but $P_{s} u\left(t_{e}\right)$ decreased to order $\mathcal{O}\left(\varepsilon^{2}\right)$.
Theorem 2.4 (Attractivity) Let Assumptions 2.1, 2.3, and 2.4 be true.
For all small $\kappa>0$, all $\delta>0$ and $p>0$ there are constants $C>0, \delta_{1}, \delta_{2}>0$ such that for $t_{e}=\frac{2}{\omega} \ln \left(\varepsilon^{-1}\right)$ and all mild solutions in the sense of Definition $2.5$
$$\mathbb{P}\left(\left|u\left(t_{\varepsilon}\right)\right| \leq \delta_{1} \varepsilon,\left|P_{s} u\left(t_{\varepsilon}\right)\right| \leq \delta_{2} \varepsilon^{2}\right) \geq \mathbb{P}(|u(0)| \leq \delta \varepsilon)-C \varepsilon^{p}$$
for all $\varepsilon \in(0,1)$.
The proof relies on the linear stability of (2.40) and cut-off techniques. We postpone the proof to Section $2.4 .4$ as it is not difficult but technical. Let us first discuss the results for the residual and the approximation.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Residual

For a solution of the amplitude equation $(2.45)$ and some $\psi(0)$, we consider the approximation $\varepsilon w$ given by $(2.47)$. The residual of $\varepsilon w$ is as usual defined as
\begin{aligned} \operatorname{Res}(\varepsilon w)(t)=&-\varepsilon w(t)+\varepsilon \mathrm{e}^{t L} w(0)+\varepsilon^{2} \int_{0}^{t} \mathrm{e}^{(t-\tau) L} w(\tau) d \beta(\tau) \ &+\int_{0}^{t} \mathrm{e}^{(t-\tau) L}\left\varepsilon^{3} A w+\varepsilon^{2} B(w)\right d \tau \end{aligned}
Theorem 2.5 (Residual) Let Assumptions 2.1, 2.3, and 2.4 be true.
For $p>4, \delta>0, T_{0}>0$ there is a constant $C>0$ such that for all approximations defined by (2.46) and (2.47), where a is a solution of (2.45), with $\mathbb{E}|a(0)|^{4 p} \leq \delta$ and $\mathbb{E}|\psi(0)|^{2 p} \leq \delta$ we have
$$\mathbb{E}\left(\sup {t \in\left[0, T{0} \varepsilon^{-2}\right]}\left|P_{c} \operatorname{Res}(\varepsilon w)(t)\right|^{p}\right) \leq C \varepsilon^{2 p}$$
and
$$\mathbb{E}\left(\sup {t \in\left[0, T{0} \epsilon^{-2}\right]}\left|P_{s} \operatorname{Res}(\varepsilon w)(t)\right|^{p}\right) \leq C \varepsilon^{3 p-2}$$
Furthermore $P_{c} \operatorname{Res}(\varepsilon w)$ is differentiable with
$$\partial_{t} P_{c} \operatorname{Res}(\varepsilon w)(t)=\varepsilon^{4}\left[A_{c} \psi+B_{c}(\psi)\right]\left(\varepsilon^{2} t\right)$$
The proof below is straightforward using the key Lemma 2.2. This lemma is a purely technical estimate, and we postpone the proof to Section 2.4.4.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Results for Quadratic Nonlinearities

∂吨在=大号在+e2一种在+乙(在,在)+e在b˙,

∂吨一种=一种C一种−2乙C(一种,大号s−1乙s(一种,一种))+一种∂吨b~

$$a(T)=a(0)+\int_{0}^{T}\left A_{c} a-2 B_{c}\left(a, L_{s}^{- 1} B_{s}(a, a)\right)\right d \tau+\int_{0}^{T} a(\tau) d \tilde{\beta}(\tau)$$

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Residual

\begin{aligned} \operatorname{Res}(\varepsilon w)(t)=&-\varepsilon w(t)+\varepsilon \mathrm{e}^{t L} w(0 )+\varepsilon^{2} \int_{0}^{t} \mathrm{e}^{(t-\tau) L} w(\tau) d \beta(\tau) \ &+\int_{ 0}^{t} \mathrm{e}^{(t-\tau) L}\left \varepsilon^{3} A w+\varepsilon^{2} B(w)\right d \tau \end{对齐} 吨H和这r和米2.5(R和s一世d在一种l)大号和吨一种ss在米p吨一世这ns2.1,2.3,一种nd2.4b和吨r在和.F这rp>4,d>0,吨0>0吨H和r和一世s一种C这ns吨一种n吨C>0s在CH吨H一种吨F这r一种ll一种ppr这X一世米一种吨一世这nsd和F一世n和db是(2.46)一种nd(2.47),在H和r和一种一世s一种s这l在吨一世这n这F(2.45),在一世吨H和|一种(0)|4p≤d一种nd和|ψ(0)|2p≤d在和H一种在和 \mathbb{E}\left(\sup {t \in\left[0, T{0} \varepsilon^{-2}\right]}\left|P_{c} \operatorname{Res}(\varepsilon w )(t)\right|^{p}\right) \leq C \varepsilon^{2 p} 一种nd \mathbb{E}\left(\sup {t \in\left[0, T{0} \epsilon^{-2}\right]}\left|P_{s} \operatorname{Res}(\varepsilon w )(t)\right|^{p}\right) \leq C \varepsilon^{3 p-2} F在r吨H和r米这r和磷C水库⁡(e在)一世sd一世FF和r和n吨一世一种bl和在一世吨H \partial_{t} P_{c} \operatorname{Res}(\varepsilon w)(t)=\varepsilon^{4}\left[A_{c} \psi+B_{c}(\psi)\right] \left(\varepsilon^{2} t\right)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|随机偏微分方程代写Stochastic partialResidual

$$\partial_{t} u=\Delta u+\xi,$$

statistics-lab™ 为您的留学生涯保驾护航 在代写随机偏微分方程方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机偏微分方程代写方面经验极为丰富，各种代写随机偏微分方程相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Residual

With Theorem $2.1$ at hand we make the following ansatz
$$u(t)=\varepsilon a\left(\varepsilon^{2} t\right)+\mathcal{O}\left(\varepsilon^{2}\right), \quad \text { where } a \in \mathcal{N} .$$
Using a formal calculation completely analogous to the one of Section $1.1 .1$ yields in lowest order of $\varepsilon>0$ the following amplitude equation:
$$d a=A_{c} a+\mathcal{F}{c}(a)+a d \tilde{\beta},$$ where ${\bar{\beta}(T)}{T \geq 0}$ defined by $\bar{\beta}(T)=\varepsilon \beta\left(\varepsilon^{-2} T\right)$ is a rescaled version of the Brownian motion $\beta$. As usual we consider the equation in the Itô sense. Note again, as explained in Section 1.1.1, that a fixed realization of the amplitude equation obviously depends on $\varepsilon$, but in distribution the solutions are independent of $\varepsilon$.
For a solution $a$ of (2.15) we define the residual
\begin{aligned} \operatorname{Res}(\varepsilon a)\left(\varepsilon^{2} t\right)=-\varepsilon a\left(\varepsilon^{2} t\right) &+\varepsilon \mathrm{e}^{t L} a(0)+\varepsilon^{2} \int_{0}^{t} \mathrm{e}^{(t-\tau) L} a\left(\varepsilon^{2} \tau\right) d \beta(\tau) \ &+\varepsilon^{3} \int_{0}^{t} \mathrm{e}^{(t-\tau) L}[A a+\mathcal{F}(a)]\left(\varepsilon^{2} \tau\right) d \tau \end{aligned}
We show:
Theorem 2.2 (Residual) Let Assumptions 2.1, 2.2, and 2.3 be true. Then for all $p>\frac{4}{3}, \delta>0$ and $T_{0}>0$ there is a constant $C>0$ such that
$$\mathrm{P}{\mathrm{c}} \operatorname{Res}(\varepsilon a)\left(\varepsilon^{2} t\right)=0$$ and $$\mathbb{E}\left(\sup {t \in\left[0, T_{0} \varepsilon^{-2}\right]}\left|P_{s} \operatorname{Res}(\varepsilon a)\left(\varepsilon^{2} t\right)\right|^{p}\right) \leq C \varepsilon^{3 p}$$
for all sufficiently small $\varepsilon>0$ and all solutions a of (2.15) with $\mathbb{E}|a(0)|^{3 p} \leq \delta \varepsilon^{3 p}$.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Approximation

Define the remainder $R$, which is the error of our approximation, as
$$\varepsilon^{2} R(t)=u(t)-\varepsilon a\left(\varepsilon^{2} t\right)$$
We split
$$R=R_{c}+R_{s} \quad \text { with } \quad R_{c}=P_{c} R \text { and } R_{s}=P_{s} R .$$
First we treat $R_{s}$ using the a priori estimates on $P_{s} u$. This information on $P_{s} u$ is not necessary for the result, as we can use cut-off techniques to yield local results, but here it helps to simplify the proofs a lot. The a priori estimates on $u$ are only possible because of the very strong stability assumptions on $\mathcal{F}$. Our main result is the following:

Theorem 2.3 (Approximation) Let Assumptions 2.1, 2.2, and $2.3$ be true. For $p>4, T_{0}>0$, and $\delta>0$ there is a constant $C>0$ such that for all strong solutions $u$ of (2.3) in $X$ with
$$\mathbb{E}|u(0)|^{3 p} \leq \delta \varepsilon^{3 p} \quad \text { and } \quad \mathbb{E}\left|P_{s} u(0)\right|^{p} \leq \delta \varepsilon^{3 p}$$
for all $\varepsilon \in(0,1)$, we derive
$$\left.\mathbb{E}\left(\sup {t \in\left[0, T{0} e^{-2}\right]} | P_{s} R(t)\right) |^{p}\right) \leq C \varepsilon^{p}$$
and
$$\left.\mathbb{E}\left(\sup {t \in\left[0, T{\mathrm{b}} e^{-2}\right]} | P_{c} R(t)\right) |^{p}\right) \leq C$$
for all sufficiently small $\varepsilon>0$, where $a$ is a solution of (2.15) such that $a(0)=$ $\varepsilon^{-1} P_{c} u(0)$.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|A priori Estimates for u

The following section provides standard a priori estimates for solutions of (2.3). Although they are straightforward, they are nevertheless quite technical. We establish bounds for $\mathbb{E}|u(t)|^{p}$ and $\mathbb{E}\left|P_{s} u(t)\right|^{p}$, which are used in the proof of Theorem 2.1. Furthermore, we bound $\mathbb{E} \sup {t \in\left[0, T{0} e^{-2}\right]}|u(t)|^{p}$ and in Lemma $2.1$ $\mathbb{E} \sup {t \in\left[0, T{\mathrm{b}} e^{-2}\right]}\left|P_{s} u(t)\right|^{p}$. The main idea is to apply Itô’s formula to $|u(t)|^{p}$ and to use the strong nonlinear stability condition from (2.7). The main technical obstacle is that a priori we do not know that $\mathbb{E}|u(t)|^{p}$ exists. Therefore we use cut-off techniques.

Proof. (of Theorem 2.1) For $p \geq 2$ and $\gamma>0$ consider smooth bounded $\varphi_{\gamma, p}:[0, \infty) \rightarrow \mathbb{R}$ such that $0 \leq \varphi_{\gamma, p}(z) \nearrow \varphi_{p}(z)=z^{p / 2}$. To be more precise, define
$$\varphi_{\gamma, p}(z):=\left(\frac{z}{1+\gamma z}\right)^{p / 2} \text { for } z \geq 0 .$$
It is now easy to check that there are constants $C_{p}$ and $c_{p}$ independent of $\gamma$ such that for $z \geq 0$
$$\begin{gathered} 0 \leq \varphi_{\gamma, p}^{\prime}(z) z \leq C_{p} \varphi_{\gamma, p}(z), \quad-p \varphi_{\gamma, p}(z) \leq \varphi_{\gamma, p}^{\prime \prime}(z) z^{2} \leq C_{p} \varphi_{\gamma, p}(z), \ \varphi_{\gamma, p}^{\prime}(z) z^{2}=\frac{p}{2} \varphi_{\gamma, p}(z)^{(p+2) / p}, \quad \varphi_{\gamma, p}^{\prime}(z) z^{2} \leq \frac{p}{2} \varphi_{\gamma, p-2}(z)=\frac{p}{2} \varphi_{\gamma, p}(z)^{(p-2) / p} . \end{gathered}$$
Apply Itô’s formula to $\varphi_{\gamma, p}\left(|u(t)|^{2}\right)$ for $t<\tau_{e}$ to derive
$$\begin{gathered} d \varphi_{\gamma, p}\left(|u(t)|^{2}\right)=\varphi_{\gamma, p}^{\prime}\left(|u(t)|^{2}\right)\left\langle u(t), L u(t)+\varepsilon^{2} A u(t)+\mathcal{F}(u(t))\right\rangle d t \ +\varphi_{\gamma, p}^{\prime}\left(|u(t)|^{2}\right)|u(t)|^{2}\left[\varepsilon d \beta(t)+\frac{1}{2} \varepsilon^{2} d t\right] \ +\varphi_{\gamma, p}^{\prime \prime}\left(|u(t)|^{2}\right)|u(t)|^{4} \varepsilon^{2} d t . \end{gathered}$$
Hence, for $t<\tau_{0}$ as we are dealing with strong solutions in the sense of Definition $2.4$.
\begin{aligned} &\mathbb{E} \varphi_{\gamma, p}\left(|u(t)|^{2}\right)-\mathbb{E} \varphi_{\gamma, p}\left(|u(0)|^{2}\right) \ &=\int_{0}^{t} \mathbb{E} \varphi_{\gamma, p}^{\prime}\left(|u(\tau)|^{2}\right)\left(u(\tau), L u(\tau)+\varepsilon^{2} A u(\tau)+\mathcal{F}(u(\tau))\right\rangle d \tau \ &\quad+\frac{1}{2} \varepsilon^{2} \int_{0}^{t} \mathbb{E} \varphi_{\gamma, p}^{\prime}\left(|u(\tau)|^{2}\right)|u(\tau)|^{2} d \tau \ &+\varepsilon^{2} \int_{0}^{t} \mathbb{E} \varphi_{\gamma, p}^{\prime \prime}\left(|u(\tau)|^{2}\right)|u(\tau)|^{4} d \tau \end{aligned}

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Residual

u(t)=\varepsilon a\left(\varepsilon^{2} t\right)+\mathcal{O}\left(\varepsilon^{2}\right)， \quad \text { 其中 } 一个 \in \mathcal{N} 。使用与第1.1 节 .1u(t)=εa(ε2t)+O(ε2), where a∈N.

Res⁡(εa)(ε2t)=−εa(ε2t)+εetLa(0)+ε2∫0te(t−τ)La(ε2τ)dβ(τ) +ε3∫0te(t−τ)L[Aa+F(a)](ε2τ)dτ

PcRes⁡(εa)(ε2t)=0和E(supt∈[0,T0ε−2]|PsRes⁡(εa)(ε2t)|p)≤Cε3p

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Approximation

ε2R(t)=u(t)−εa(ε2t)

R=Rc+Rs with Rc=PcR and Rs=PsR.

E|u(0)|3p≤δε3p and E|Psu(0)|p≤δε3p
ε∈(0,1)
E(supt∈[0,T0e−2]|PsR(t))|p)≤Cεp

E(supt∈[0,Tbe−2]|PcR(t))|p)≤C

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|A priori Estimates for u

φγ,p(z):=(z1+γz)p/2 for z≥0.

0≤φγ,p′(z)z≤Cpφγ,p(z),−pφγ,p(z)≤φγ,p′′(z)z2≤Cpφγ,p(z), φγ,p′(z)z2=p2φγ,p(z)(p+2)/p,φγ,p′(z)z2≤p2φγ,p−2(z)=p2φγ,p(z)(p−2)/p.

dφγ,p(|u(t)|2)=φγ,p′(|u(t)|2)⟨u(t),Lu(t)+ε2Au(t)+F(u(t))⟩dt +φγ,p′(|u(t)|2)|u(t)|2[εdβ(t)+12ε2dt] +φγ,p′′(|u(t)|2)|u(t)|4ε2dt.

Eφγ,p(|u(t)|2)−Eφγ,p(|u(0)|2) =∫0tEφγ,p′(|u(τ)|2)(u(τ),Lu(τ)+ε2Au(τ)+F(u(τ))⟩dτ +12ε2∫0tEφγ,p′(|u(τ)|2)|u(τ)|2dτ +ε2∫0tEφγ,p′′(|u(τ)|2)|u(τ)|4dτ

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|随机偏微分方程代写Stochastic partial differentialAmplitude Equations on Bounded Domains

$$\partial_{t} u=\Delta u+\xi,$$

statistics-lab™ 为您的留学生涯保驾护航 在代写随机偏微分方程方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机偏微分方程代写方面经验极为丰富，各种代写随机偏微分方程相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Multiplicative Noise

Let us first motivate, why we are interested in multiplicative noise. It appears naturally in models, where one considers noisy control parameters. Consider as an example some deterministic PDE of the following type
where $L$ is some linear differential operator and $\mathcal{F}$ is some nonlinearity, for instance $-u^{3}$. Suppose that the equation undergoes a change of stability (or bifurcation) when $\mu=0$.

The question is, whether we can see the influence of small noise in the bifurcation parameter $\mu$ in the case where $\mu$ is near or at the bifurcation. This is an important question in many experiments, as $\mu$ models experimental quantities like, for instance, temperature, which are naturally subject to small (random) perturbations.

We consider in (2.1) a simplified PDE model, where the perturbation of the parameter has no spatial dependence and is homogeneous in space. This kind of equation was recently studied in more detail, for instance, by [CLR00; CLR01; Rob02] where they determined the dimension and structure of a random attractor for a stochastic Ginzburg-Landau equation. On the other hand, even the stability of linear equations (i.e. $\mathcal{F} \equiv 0$ ) was only studied recently in [CR04] or [Kwi02] following the celebrated work of [ACW83].

Let us come back to (2.1). Assume that the control parameter $\mu \in \mathbb{R}$ is perturbed by white noise and suppose the strength of the fluctuations $\varepsilon>0$ is small. A typical model is a Gaussian noise $\mu$ with some mean and covariance functional
$$\mathbb{E} \mu(t)=\mu_{\varepsilon} \in \mathbb{R}, \quad \mathbb{E}\left(\mu(t)-\mu_{\varepsilon}\right)\left(\mu(s)-\mu_{\varepsilon}\right)=\varepsilon^{2} \delta(t-s) .$$
Thus we can write $\mu=\mu_{\varepsilon}+\varepsilon \xi$, where $\xi=\partial_{t} \beta$ is the generalised derivative of a real valued Brownian motion $\beta={\beta(t)}_{t \geq 0}$.
Hence, we can rewrite (2.1) as a stochastic PDE
$$\partial_{t} u=L u+\mu_{\varepsilon} u+\mathcal{F}(u)+\varepsilon u \partial_{t} \beta$$

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Assumptions and Results The Cubic Case

This section summarises for the cubic case all assumptions necessary and states the main results. In this chapter we treat two sets of different assumptions. On one hand this section treats nonlinear stable equations involving cubic terms, where we can use standard a priori estimates to obtain bounds on moments of solutions. On the other hand we consider in Section $2.4$ quadratic nonlinearities, which in general do not allow to bound moments of solutions. Especially, if we cannot rule out the possibility of a blow-up of solutions in finite time, which is the case in many examples. One is the 2D Kuramoto-Sivashinsky equation, for instance. In this case we obtain local result by using cut-off techniques.

Consider the following SPDE in some Hilbert space $X$ with scalar product $\langle\cdot,\rangle$, and norm $|\cdot|$. We could also consider Banach spaces here, but the Hilbert space setting simplifies the notation and the a priori estimates on solutions.
$$d u=\left[L u+\varepsilon^{2} A u+\mathcal{F}(u)\right] d t+\varepsilon u d \beta$$
The precise setting is given below in Assumptions $2.1$ for $L, 2.2$ for $A$ and $\mathcal{F}$, and $2.3$ for the Itô-differential.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Attractivity

We establish two results. The first one in Theorem $2.1$ is a very strong result. It relies on the nonlinear stability of the equation and establishes bounds on $\mathbb{E}|u(t)|^{p}$ for large $t$ completely independent of the initial condition $u(0)$. The second result is somewhat weaker. It relies on the existence of bounds on $\mathbb{E}|u(0)|^{p}$, and it establishes bounds on $\mathbb{E}\left|P_{s} u(t)\right|^{p}$ for moderately large $t$. This relies mainly on the linearised picture and a spectral gap of the linearised operator.

For the attractivity our main goal is to verify that there is a time $t_{\varepsilon}>0$ such that
$$u\left(t_{e}\right)=\varepsilon a_{e}+\varepsilon^{3} \psi_{e},$$
where $a_{\varepsilon} \in \mathcal{N}$ and $\psi_{e} \in P_{s} X$ are both of order $\mathcal{O}(1)$.
Theorem 2.1 (Attractivity) Let Assumptions 2.1, 2.2, and $2.3$ be true and let $u$ be a strong solution of (2.3) in $X$.
Then for all $p>0$ and $t_{0}>0$ there is a constant $C>0$ such that
$$\sup {t \geq t{0} e^{-2}} \mathbb{E}|u(t)|^{p} \leq C \varepsilon^{p}$$
for all sufficiently small $\varepsilon>0$ and all strong solutions $u$ of (2.3) in $X$ independent of the initial condition. Especially, $\tau_{e}=\infty$ almost surely for the maximal time of existence of $u$.

Furthermore, for $q \geq 2, \delta>0$, and $p \in[2, q]$ there is some constant $C>0$ such

that $\mathbb{E}|u(0)|^{q} \leq \delta \varepsilon^{q}$ for all $\varepsilon \in(0,1)$ implies
$$\sup {t \geq 0} \mathbb{E}|u(t)|^{p} \leq C \varepsilon^{p} \quad \text { for all sufficiently small } \varepsilon>0 \text {. }$$ Additionally, for $t{e}=\frac{2}{\omega} \ln \left(\varepsilon^{-1}\right)$ and all $p \in[4, q / 3]$ there is a constant $C>0$ such that
$$\sup {t \geq t{\varepsilon}} \mathbb{E}\left|P_{s} u(t)\right|^{p} \leq C \varepsilon^{3 p} \quad \text { for all sufficiently small } \varepsilon>0 .$$
The proof is straightforward. But, as it is quite technical, we postpone it to Section 2.3. The main tools are standard a priori type estimates using Itô’s formula and Burkholder’s inequality.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Multiplicative Noise

∂吨在=大号在+μe在+F(在)+e在∂吨b

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Assumptions and Results The Cubic Case

d在=[大号在+e2一种在+F(在)]d吨+e在db

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|随机偏微分方程代写Stochastic partial differentialMeta Theorems

$$\partial_{t} u=\Delta u+\xi,$$

statistics-lab™ 为您的留学生涯保驾护航 在代写随机偏微分方程方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机偏微分方程代写方面经验极为丰富，各种代写随机偏微分方程相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Meta Theorems

The first result presented here is the attractivity. It justifies the scaling of ansatz (1.4) used for the formal derivation. It heavily relies on the structure of the equation. Sometimes we rely on global nonlinear stability and sometimes we only use linear stability on the non-dominant modes. A typical statement would be:

Theorem 1.1 (Attractivity) There is a time $t_{e}=\mathcal{O}\left(\ln \left(\varepsilon^{-1}\right)\right)$ such that for all solutions $u$ of (1.19) with initial conditions $u(0)$ of order $\mathcal{O}(\varepsilon)$ we have $u_{s}\left(t_{e}\right)=$ $\mathcal{O}\left(\varepsilon^{2}\right)$ and $u_{c}\left(t_{e}\right)=\mathcal{O}(\varepsilon)$. This means the solution looks at the time $t_{e}$ like ansatz (1.4). To be more precise $u\left(t_{\varepsilon}\right)=\varepsilon a_{e}+\varepsilon^{2} \psi_{\varepsilon}$ with $a_{\varepsilon} \in \mathcal{N}$ and $\psi_{e} \in \mathcal{S}$ both of order $\mathcal{O}(1)$.

If we assume additionally global nonlinear stability for the equation, then there is a time $T_{\varepsilon}=\mathcal{O}\left(\varepsilon^{-2}\right)$ such that $u\left(T_{\varepsilon}\right)=\mathcal{O}(\varepsilon)$ independent of the initial condition.
This theorem is rigorously stated in Theorems $2.7$ or $2.8$. We will give a detailed discussion of these results for multiplicative noise in Theorems $2.1$ and $2.4$ for cubic and quadratic nonlinearities. A sketch of the typical dynamics for the local attractivity result is given in Figure 1.3.

Remark 1.4 Depending on the assumptions the statement $g_{e}=\mathcal{O}\left(f_{\varepsilon}\right)$ can have two different meanings. Depending on the context, we either use that for all $p>0$ there is a constant $C>0$ such that $\mathbb{E}\left|g_{e}\right|^{p} \leq C f_{\varepsilon}^{p}$ for all $\varepsilon \in(0,1]$. This is typically

only valid for nonlinear stable equations, where we can actually bound moments. In case of, for instance quadratic nonlinearities, where in general we do not have control on moments of solutions, we also use the somewhat weaker meaning that for some constant $C>0$, we have $\mathbb{P}\left(\left|g_{\varepsilon}\right| \geq C f_{\varepsilon}\right)$ uniformly small for all $\varepsilon \in(0,1]$. Sometimes we also give explicit convergence rates of this probability for $\varepsilon \rightarrow 0$.
For a solution $a$ of $(1.5)$ and $\psi$ of (1.6) we define first the approximations $\varepsilon w_{k}$ of order $k$ by
$$\varepsilon w_{1}(t):=\varepsilon a\left(\varepsilon^{2} t\right) \text { and } \varepsilon w_{2}(t):=\varepsilon a\left(\varepsilon^{2} t\right)+\varepsilon^{2} \psi(t)$$
In our setting the residual of $\varepsilon w$ is defined by
\begin{aligned} \operatorname{Res}(\varepsilon w)(t)=&-\varepsilon w(t)+\mathrm{e}^{t L} \varepsilon w(0)+\varepsilon^{2} W_{L}(t) \ &+\int_{0}^{t} \mathrm{e}^{(t-\tau) L}\left\varepsilon^{3} A w+\mathcal{F}(\varepsilon w)\right d \tau \end{aligned}

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Examples of Equations

In the literature there are numerous examples of equations where the abstract theorems do apply. In this section we focus mainly on additive noise. For instance, for cubic nonlinearities the well known Ginzburg-Landau equation (see [DE00] for a standard proof of existence of unique solutions)
$$\partial_{t} u=\Delta u+\nu u-u^{3}+\sigma \xi$$
and the Swift-Hohenberg equation (see [CH93] for numerous references)
$$\partial_{t} u=-(\Delta+1)^{2} u+\nu u-u^{3}+\sigma \xi$$

fall into the scope of our work, in case the parameters $\nu$ and $\sigma$ are small and of comparable order of magnitude. Both equations are considered on bounded domains with suitable boundary conditions (e.g. periodic, Dirichlet, Neumann, etc.). The Swift-Hohenberg equation is a toy model for the convective instability in the Rayleigh-Bénard convection. A formal derivation of the equation from the Boussinesq approximation of fluid dynamics can be found in [SH77].
Another example arising in the theory of surface growth is
$$\partial_{t} u=-\Delta^{2} u-\mu \Delta u+\nabla \cdot\left(|\nabla u|^{2} \nabla u\right)+\sigma \xi,$$
subject to periodic boundary conditions and moving frame $\int_{G} u d x=0$, where one rescales the mean growth of $u$ out of the equation, in order to ensure a Poincare type inequality. This model was first suggested by [LDS91]. The deterministic equation was rigorously treated in [KSW03]. For a review on surface growth see for example [BS95] or [HHZ95]. For this model we can consider $\mu=\mu_{0}+\varepsilon^{2}$ and $\sigma=\mathcal{O}\left(\varepsilon^{2}\right)$,where $\mu_{0}$ is such that $L=-\Delta^{2} u-\mu_{0} \Delta u$ is a non-positive operator with non-zero kernel. We will see later on, that all examples presented up to now exhibit a stable nonlinearity in the sense of Assumption 2.2.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Bounded Domains

On bounded domains, we can approximate on long time-scales the essential dynamics of an SPDE near a change of stability by the amplitude equation. This is in this chapter just an SDE describing the dynamics of the dominating modes, which are the ones that change sign in the linearisation. For the formal derivation in the case of additive noise see Sections 1.1.1 or 1.1.3. The main mathematical reason why the other modes are not important is the presence of a well defined spectral gap in the linearised equation of order $\mathcal{O}(1)$ between the eigenvalues of the dominant eigenfunctions and the remainder of the spectrum.

The approximation via SDE is only meaningful for small domains. If the domain gets larger, one needs very small noise to apply the results. See Chapter 4 , where the size of the domain is coupled to the distance from bifurcation. Problems arise due to the fact that, if we enlarge the domain, then we shrink the spectral gap. The precise definition of the spectral gap $\omega$ will be given in Assumption 2.1. The main problem is that a lot of constants depend on $\omega$, and they tend to infinity for $\omega \rightarrow 0$. But if the domain-size $\ell \rightarrow \infty$, then in most cases $\omega \rightarrow 0$. Hence, for large domains our result is only meaningful for very small noise strength $\varepsilon^{2}$ with $\varepsilon \in\left(0, \varepsilon_{0}\right]$, where $\varepsilon_{0}=\varepsilon_{0}(\ell) \rightarrow 0$ for $\ell \rightarrow \infty$. However, the linear part of our equation is usually coupled to the noise, and thus has to be very small, too. The main problem is now, that this linear part reflects the influence of control parameters adjusted in experiments. It is not possible to consider it arbitrarily small.

In the following, we demonstrate the power of our approach by applying it to PDEs perturbed by a simple multiplicative noise. Although our results apply to more complicated noise terms, for simplicity of presentation we consider only this very simple example in order to outline the main ideas in a less technical way. The results for additive noise are reviewed later in this chapter.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Meta Theorems

e在1(吨):=e一种(e2吨) 和 e在2(吨):=e一种(e2吨)+e2ψ(吨)

\begin{aligned} \operatorname{Res}(\varepsilon w)(t)=&-\varepsilon w(t)+\mathrm{e}^{t L} \varepsilon w(0)+ \varepsilon^{2} W_{L}(t) \ &+\int_{0}^{t} \mathrm{e}^{(t-\tau) L}\left \varepsilon^{3} A w+ \mathcal{F}(\varepsilon w)\right d \tau \end{aligned}

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Examples of Equations

∂吨在=Δ在+ν在−在3+σX

∂吨在=−(Δ+1)2在+ν在−在3+σX

∂吨在=−Δ2在−μΔ在+∇⋅(|∇在|2∇在)+σX,

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Quadratic Nonlinearities

$$\partial_{t} u=\Delta u+\xi,$$

statistics-lab™ 为您的留学生涯保驾护航 在代写随机偏微分方程方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机偏微分方程代写方面经验极为丰富，各种代写随机偏微分方程相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Quadratic Nonlinearities

An interesting feature of quadratic nonlinearities $B(u)=B(u, u)$ is that in many examples $P_{c} B(a) \equiv 0$ for all $a \in \mathcal{N}$. In this case, the ansatz (1.8) yields only the linearisation. See (1.9). This means that we still look at solutions that are too small to capture any of the nonlinear effects present in the equation. In order to obtain a nonlinear amplitude equation, we either consider larger noise, or we look at a parameter regime where we are nearer to the change of stability.

To illustrate this problem, we briefly discuss a one-dimensional Burgers’ equation, which is given by
$$\partial_{t} u=\partial_{x}^{2} u+\mu_{e} u-v \partial_{x} u+\sigma_{\varepsilon} \xi .$$
Let $\xi$ be space-time white noise for simplicity.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Large or Unbounded Domains

For unbounded domains the results are very different. First of all, we do not have a spectral gap, and near the change of stability a whole band of eigenvalues gets unstable. The same effect already occurs, if we consider large domains, which are at least of the size $\mathcal{O}\left(\varepsilon^{-1}\right)$. In Figure $1.1$ we briefly sketch the eigenvalue curve $k \mapsto-P(-k)$ with the corresponding eigenvalues of the Swift-Hohenberg operator $-P\left(i \partial_{x}\right)=-\left(1+\partial_{x}^{2}\right)^{2}$. For the deterministic PDE this somewhat intermediate step was already discussed in [MSZ00]. The stochastic case is treated in [BHP05], but we present a different formal derivation here. This is closer to usual physical reasoning, and more in the spirit of [KSM92].

Consider as an example a one-dimensional version of the Swift-Hohenberg equation, which was first used as a toy-model for the convective instability in the

Rayleigh-Bénard problem (see [SH77]). Here
$$u(t, x) \in \mathbb{R}, \quad \text { for } \quad t>0, x \in D_{\varepsilon}=L \varepsilon^{-1},[-1,1]$$
fulfils
$$\partial_{t} u=-P\left(i \partial_{x}\right) u+\varepsilon^{2} \nu u-u^{3}+\varepsilon^{\frac{3}{2}} \xi$$
subject to periodic boundary conditions. Note that we prescribe a scaling between the noise strength and the distance from bifurcation, that differs from the one used in the bounded domain case.
The linear operator is given by
$$P(\zeta)=\left(1-\zeta^{2}\right)^{2} .$$
The complex eigenfunctions of the linear operator $P\left(i \partial_{x}\right)$ are $x \mapsto \exp {i k \varepsilon \pi x / L}$ with corresponding eigenvalue $P(k \varepsilon \pi / L)$ for $k \in \mathbb{Z}$. For simplicity, let $\xi$ be spacetime white noise in the following formal calculation. We rely on scaling properties for the noise, which are not that easy to formulate for coloured noise. See also Section 4.2. To be more precise, we use that $\xi$ and $\hat{\xi}$ are versions of the same noise, when we define
$$\hat{\xi}(T, X)=\varepsilon^{-3 / 2} \xi\left(T \varepsilon^{-2}, X \varepsilon^{-1}\right)$$
We expect a linear instability at $\mathrm{e}^{\pm i x}$, as $P(\pm 1)=0$ and $P(x)>0$ for $x \neq \pm 1$, but due to the boundedness of the domain $\mathrm{e}^{\pm i x}$ is in general not an eigenfunction. The nearest eigenfunction is $\mathrm{e}^{i \rho_{c}(e / L) x}$, where
$$\rho_{c}(\varepsilon / L):=\frac{\varepsilon \pi}{L} \cdot\left[\frac{L}{\varepsilon \pi}\right]$$

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|General Structure of the Approach

It is not the aim of this section to present rigorous results. Instead it highlights the key steps in a non-technical way. For all our results in the stochastic case, the general method of proof already dates back to [BMPS01]. Furthermore it was already used for amplitude equations for deterministic equations, for instance, in [KSM92] and [Sch94].

For simplicity of presentation we focus on the case of bounded domains. The case of large or unbounded domains is similar, but it exhibits many additional technical difficulties. Furthermore, we stick to cubic nonlinearities with additive noise. This was discussed in Section 1.1.1. The method of proof for other types of equations is very similar, only the formulation and the technical details differ.

Due to the lack of regularity, we cannot proceed analogous to the deterministic setting. This is one of the main issues for SPDEs, as the approach for deterministic PDEs relies on bounds for solutions of the amplitude equations in spaces with sufficiently high regularity. But especially on large domains for SPDEs this is never the case. See Section $4.3$ or Remark 4.1.

In order to give SPDEs like (1.2) a meaning, we use the concept of mild solutions. These are stochastic processes with continuous paths that fulfil the following variation of constants formula
$$u(t)=\mathrm{e}^{t L} u(0)+\int_{0}^{t} \mathrm{e}^{(t-\tau) L}\left\varepsilon^{2} A u+\mathcal{F}(u)\right d \tau+\varepsilon^{2} W_{L}(t)$$
for $t \leq t^{}$, where $t^{}>0$ is some stopping time. Here $\left{\mathrm{e}^{t L}\right}_{t \geq 0}$ denotes the semigroup of operators generated by the differential operator $L$. For a detailed definition see [Paz83; Hen81; Lun95] or Section 2.5.1. The main point here is that $w(t)=\mathrm{e}^{t L} w_{0}$ solves $\partial_{t} w=L w$ with $w(0)=w_{0}$, and thus $\partial_{t} \mathrm{e}^{t L}=L \mathrm{e}^{t L}$.
For the definition of the stochastic convolution
$$W_{L}(t)=\int_{0}^{t} \mathrm{e}^{(t-\tau) L} d Q W(\tau), \quad t \geq 0$$

see [DPZ92]. Formally differentiating (1.19) yields immediately that $u(t)$ solves (1.2).

Here $\partial_{t} Q W=\xi$ in a generalised sense, and $W$ is some cylindrical Wiener process in some Hilbert space (see Assumption $2.8$ and the discussion below that). For the connection between the noise $\xi$ and $Q$-Wiener processes see [Blö05b]. For a different approach using the Brownian sheet and an explicit representation of the semigroup $\mathrm{e}^{t L}$ via the Green function see [Wal86].

We use the projection $P_{c}$ onto the kernel $\mathcal{N}$ of $L$ and $P_{s}=I-P_{c}$, which were defined before (cf. Section 1.1.1). Now we project the equation to $\mathcal{N}$ and $\mathcal{S}$.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Quadratic Nonlinearities

∂吨在=∂X2在+μ和在−在∂X在+σeX.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Large or Unbounded Domains

Rayleigh-Bénard 问题（参见 [SH77]）。这里

∂吨在=−磷(一世∂X)在+e2ν在−在3+e32X

X^(吨,X)=e−3/2X(吨e−2,Xe−1)

ρC(e/大号):=e圆周率大号⋅[大号e圆周率]

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|General Structure of the Approach

$$u(t)=\mathrm{e}^{t L} u(0)+\int_{0}^{t} \mathrm{e }^{(t-\tau) L}\left \varepsilon^{2} A u+\mathcal{F}(u)\right d \tau+\varepsilon^{2} W_{L}(t)$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Formal Derivation of Amplitude Equations

$$\partial_{t} u=\Delta u+\xi,$$

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• Foundations of Data Science 数据科学基础

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Formal Derivation of Amplitude Equations

In this section, we discuss the formal derivation of amplitude equations and higher order corrections. Therefore, we use multiple scale analysis to reduce the equation to the essential dynamics, which involves the expansion of all terms in a small parameter. This is well known for many examples. Here we present results described in more detail for quadratic nonlinearities in [Blö05a] and for cubic nonlinearities in [BH04]. For large domains we summarise results of [BHP05] in Section 1.1.4.
Let us consider parabolic semilinear SPDEs or systems of SPDEs perturbed by additive forcing near a change of stability. Let us suppose, that the noise is of the order of the distance from the bifurcation. The use of additive noise is mainly for simplicity of presentation, and it is not very restrictive. We comment on multiplicative noise later in several occasions in Chapter 2. A large body of the research papers are on additive noise, which we will summarise later. In this book simple multiplicative noise is used to present a self-contained introduction to the topic.
The general prototype is an equation of the type
$$\partial_{t} u=L u+\varepsilon^{2} A u+\mathcal{F}(u)+\varepsilon^{2} \xi,$$
where

• $L$ is a symmetric non-positive differential operator $\left(\text { e.g. } 1+\partial_{x}^{2}\right)^{2}$ ) with non-zero finite dimensional kernel (or null-space).
• $\varepsilon^{2} A u$ is a small (linear) deterministic perturbation,
• $\mathcal{F}$ is some nonlinearity, for instance a stable cubic like $-u^{3}$.
• $\xi=\xi(t, x)$ is a Gaussian noise in space and time
We later give examples of the noise, which is taken to be white in time and can be either white or coloured in space. To be more precise, suppose that $\xi$ is a generalised Gaussian process such that for mean and correlation
$$\mathbb{E} \xi(t, x)=0 \quad \text { and } \quad \mathbb{E} \xi(t, x) \xi(s, y)=\delta(t-s) q(x-y)$$
for some suitable spatial correlation function (or distribution) $q$. If $q$ is the Deltadistribution $\delta$, too, then we call $\xi$ space-time white noise. In this case $\xi=\partial_{t} W$ is the generalised derivative of a cylindrical Wiener-process ${W(t)}_{t \geq 0}$ in a suitable Hilbert space.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Cubic Nonlinearities

One interesting example of an equation with cubic nonlinearity is the SwiftHohenberg equation, which was first used as a toy model for the convective instability in the Rayleigh-Bénard problem (see [SH77] or Section 1.3).

On a formal level for the Swift-Hohenberg equation the derivation of the amplitude equation is well known, see for instance (4.31) or (5.11) in the comprehensive review article [CH93] and references therein. The amplitude equation for (1.3) was already treated in [BMPS01]. But here we follow the presentation from [BH04], taking into account second order corrections.
The formal SPDE is
$$\partial_{t} u=-(1+\Delta)^{2} u+\varepsilon^{2} \nu u-u^{3}+\varepsilon^{2} \partial_{t} Q W .$$
It is obviously of the type of $(1.2)$ with $L=-(1+\Delta)^{2}, A=\nu I$ for some $\nu \in[-1,1]$, and $\mathcal{F}(u)=-u^{3}$. We can for instance consider periodic boundary conditions on the domain $[0,2 \pi l]^{d}$ for dimension $d \in \mathbb{N}$ and integer length $l>0$. This is mainly for convenience to ensure that the change of stability is exactly at $\nu=0$. After slight modifications we can also treat non-integer length $l>0$ or non-squared domains.
For the formal derivation in this section we consider an equation of the type (1.2) or (1.3) and assume:

Assumption 1.1 Let ${Q W(t)}_{t \geq 0}$ be a $Q$-Wiener process. This implies especially that ${W(t)}_{t \geq 0}$ and $\left{\varepsilon W\left(\varepsilon^{-2} t\right)\right}_{t \geq 0}$ are in law the same process.
Furthermore, let $\mathcal{F}$ be cubic (i.e. $\mathcal{F}(u)=\mathcal{F}(u, u, u)$ is trilinear).

Denote the kernel (or nullspace) of $L$ by $\mathcal{N}$ and the orthogonal projection onto $\mathcal{N}$ by $P_{c}$. Define $P_{s}=I-P_{c}$.
Then we make the following ansatz:
$$u(t)=\varepsilon a\left(\varepsilon^{2} t\right)+\varepsilon^{2} b\left(\varepsilon^{2} t\right)+\varepsilon^{2} \psi(t)+\mathcal{O}\left(\varepsilon^{3}\right)$$
with $a, b \in \mathcal{N}$ and $\psi \in \mathcal{S}:=\mathcal{N}^{\perp}$ the orthogonal complement of $\mathcal{N}$ in $X$.
This ansatz is motivated by the fact that, due to the linear damping of order one in $\mathcal{S}$, the modes in $\mathcal{S}$ are expected to evolve on time scales of order one, whereas the modes in $\mathcal{N}$ are expected to evolve on much slower time scales of order $\varepsilon^{-2}$, as the linear operator is of order $\varepsilon^{2}$. This is mainly due to the fact that we have a well defined spectral gap of order $\mathcal{O}(1)$ between 0 and the first non-zero eigenvalue together with a small linear perturbation of order $\varepsilon^{2}$.

We do not use lower order terms, as we expect that small solutions stay small. Furthermore, using linear and nonlinear stability, it is possible to verify a priori estimates that rigorously verify that the typical scaling of a solution corresponds to the one prescribed by the ansatz (1.4). The statement is called the attractivity result (cf. Section 1.2).

Let us now come back to the formal derivation. Plugging the ansatz (1.4) into (1.2), rescaling to the slow time-scale $T=\varepsilon^{2} t$ and expanding in orders of $\varepsilon$, we obtain by collecting all terms of order $\varepsilon^{3}$ in $\mathcal{N}$
$$\partial_{T} a(T)=A_{c} a(T)+\mathcal{F}{c}(a(T))+\partial{T} \beta(T)$$
Here,
$$\beta(T)=\varepsilon P_{c} Q W\left(\varepsilon^{-2} T\right), \quad T \geq 0$$
is a Wiener process in $\mathcal{N}$ with law independent of $\varepsilon$, due to the scaling properties of the Wiener process. We used
$$A_{c}=P_{c} A \quad \text { and } \quad \mathcal{F}{c}=P{c} \mathcal{F}$$
for short.
This approximating equation in (1.5) is called amplitude equation, as it can by rewritten to an SDE for the amplitudes of an expansion of $a$ with respect to a basis in $\mathcal{N}$. Results like this well known for many examples in the physics and applied mathematics literature (for example [CH93, (4.31), (5.11)]). Moreover, there are numerous variants of this method. However, most of these results are non-rigorous approximations using this type of formal multi-scale analysis.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Other Types of Nonlinearities

Cubic nonlinearities are not very special, we can extend the simple idea of the previous section, using scaling and projection, to a lot of different types of nonlinearities. If we look at suitable scalings of the noise and the linear (in)stability we obtain in all cases interesting results. If we do not adapt the scaling, we either loose the linear instability or the noise in the amplitude equation.

Suppose for this section that $\mathcal{F}^{(n)}$ is some multi-linear nonlinearity, which is homogeneous of degree $n \in \mathbb{N}$ with $n \geq 2$ (i.e. for $\alpha>0, \mathcal{F}^{(n)}(\alpha u)=\alpha^{n} \mathcal{F}^{(n)}(u)$ ). Then the noise strength in the SPDE $(1.2)$ should be changed to $\sigma^{2}=\varepsilon^{(n+1) /(n-1)}$ instead of $\varepsilon^{2}$. Thus the equation reads in the interesting scaling
$$\partial_{t} u=L u+\varepsilon^{2} A u+\mathcal{F}^{(n)}(u)+\varepsilon^{(n+1) /(n-1)} \xi .$$

Now with the ansatz
$$u(t)=\varepsilon^{2 /(n-1)} a\left(\varepsilon^{2} t\right)+\varepsilon^{(n+1) /(n-1)} \psi(t)+\mathcal{O}\left(\varepsilon^{2 n /(n-1)}\right)$$
and a similar formal calculation as in the previous section, we derive the same type of amplitude equation. First collecting all terms of order $\varepsilon^{2 n /(n-1)}$ in $\mathcal{N}$ yields
$$\partial_{T} a=P_{c} A a+P_{c} \mathcal{F}^{(n)}(a)+\partial_{T} \beta$$
which now contains a nonlinearity which is homogeneous of degree $n$. The second order correction is exactly the same (cf. $(1.6))$ as in the cubic case, but now it contains all terms in $\mathcal{S}$ of order $\varepsilon^{(n+1) /(n-1)}$.

We will not focus on rigorous results for this type of equations, as they are very similar to the cubic case. After minor modifications one can easily transfer all results to the general case.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Formal Derivation of Amplitude Equations

∂吨在=大号在+e2一种在+F(在)+e2X,

• 大号是一个对称的非正微分算子( 例如 1+∂X2)2) 具有非零有限维内核（或零空间）。
• e2一种在是一个小的（线性）确定性扰动，
• F是一些非线性，例如像一个稳定的立方−在3.
• X=X(吨,X)是空间和时间上的高斯噪声
我们稍后给出噪声的例子，它在时间上被认为是白色的，在空间上可以是白色或彩色的。更准确地说，假设X是一个广义高斯过程，使得对于均值和相关
和X(吨,X)=0 和 和X(吨,X)X(s,是)=d(吨−s)q(X−是)
对于一些合适的空间相关函数（或分布）q. 如果q是 Delta 分布d, 那么我们调用X时空白噪声。在这种情况下X=∂吨在是圆柱维纳过程的广义导数在(吨)吨≥0在合适的希尔伯特空间中。

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Cubic Nonlinearities

∂吨在=−(1+Δ)2在+e2ν在−在3+e2∂吨问在.

$$\partial_{T} a(T)=A_{c} a(T)+\mathcal{F} {c}(a(T))+\partial {T} \beta(T) H和r和, \beta(T)=\varepsilon P_{c} QW\left(\varepsilon^{-2} T\right), \quad T \geq 0 一世s一种在一世和n和rpr这C和ss一世nñ在一世吨Hl一种在一世nd和p和nd和n吨这Fe,d在和吨这吨H和sC一种l一世nGpr这p和r吨一世和s这F吨H和在一世和n和rpr这C和ss.在和在s和d A_{c}=P_{c} A \quad \text { 和 } \quad \mathcal{F} {c}=P {c} \mathcal{F}$$

（1.5）中的这个近似方程称为幅度方程，因为它可以重写为 SDE 的展开的幅度一种关于基础ñ. 这样的结果在物理学和应用数学文献中的许多例子中众所周知（例如 [CH93, (4.31), (5.11)]）。此外，这种方法有许多变体。然而，这些结果中的大多数是使用这种形式的多尺度分析的非严格近似。

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Other Types of Nonlinearities

∂吨在=大号在+e2一种在+F(n)(在)+e(n+1)/(n−1)X.

∂吨一种=磷C一种一种+磷CF(n)(一种)+∂吨b

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。