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

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

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

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