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

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

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## 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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