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

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

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## 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。

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