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

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

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

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