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

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

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

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