金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Formal Derivation of Amplitude Equations

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

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金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Formal Derivation of Amplitude Equations

In this section, we discuss the formal derivation of amplitude equations and higher order corrections. Therefore, we use multiple scale analysis to reduce the equation to the essential dynamics, which involves the expansion of all terms in a small parameter. This is well known for many examples. Here we present results described in more detail for quadratic nonlinearities in [Blö05a] and for cubic nonlinearities in [BH04]. For large domains we summarise results of [BHP05] in Section 1.1.4.
Let us consider parabolic semilinear SPDEs or systems of SPDEs perturbed by additive forcing near a change of stability. Let us suppose, that the noise is of the order of the distance from the bifurcation. The use of additive noise is mainly for simplicity of presentation, and it is not very restrictive. We comment on multiplicative noise later in several occasions in Chapter 2. A large body of the research papers are on additive noise, which we will summarise later. In this book simple multiplicative noise is used to present a self-contained introduction to the topic.
The general prototype is an equation of the type
$$\partial_{t} u=L u+\varepsilon^{2} A u+\mathcal{F}(u)+\varepsilon^{2} \xi,$$
where

• $L$ is a symmetric non-positive differential operator $\left(\text { e.g. } 1+\partial_{x}^{2}\right)^{2}$ ) with non-zero finite dimensional kernel (or null-space).
• $\varepsilon^{2} A u$ is a small (linear) deterministic perturbation,
• $\mathcal{F}$ is some nonlinearity, for instance a stable cubic like $-u^{3}$.
• $\xi=\xi(t, x)$ is a Gaussian noise in space and time
We later give examples of the noise, which is taken to be white in time and can be either white or coloured in space. To be more precise, suppose that $\xi$ is a generalised Gaussian process such that for mean and correlation
$$\mathbb{E} \xi(t, x)=0 \quad \text { and } \quad \mathbb{E} \xi(t, x) \xi(s, y)=\delta(t-s) q(x-y)$$
for some suitable spatial correlation function (or distribution) $q$. If $q$ is the Deltadistribution $\delta$, too, then we call $\xi$ space-time white noise. In this case $\xi=\partial_{t} W$ is the generalised derivative of a cylindrical Wiener-process ${W(t)}_{t \geq 0}$ in a suitable Hilbert space.

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

One interesting example of an equation with cubic nonlinearity is the SwiftHohenberg equation, which was first used as a toy model for the convective instability in the Rayleigh-Bénard problem (see [SH77] or Section 1.3).

On a formal level for the Swift-Hohenberg equation the derivation of the amplitude equation is well known, see for instance (4.31) or (5.11) in the comprehensive review article [CH93] and references therein. The amplitude equation for (1.3) was already treated in [BMPS01]. But here we follow the presentation from [BH04], taking into account second order corrections.
The formal SPDE is
$$\partial_{t} u=-(1+\Delta)^{2} u+\varepsilon^{2} \nu u-u^{3}+\varepsilon^{2} \partial_{t} Q W .$$
It is obviously of the type of $(1.2)$ with $L=-(1+\Delta)^{2}, A=\nu I$ for some $\nu \in[-1,1]$, and $\mathcal{F}(u)=-u^{3}$. We can for instance consider periodic boundary conditions on the domain $[0,2 \pi l]^{d}$ for dimension $d \in \mathbb{N}$ and integer length $l>0$. This is mainly for convenience to ensure that the change of stability is exactly at $\nu=0$. After slight modifications we can also treat non-integer length $l>0$ or non-squared domains.
For the formal derivation in this section we consider an equation of the type (1.2) or (1.3) and assume:

Assumption 1.1 Let ${Q W(t)}_{t \geq 0}$ be a $Q$-Wiener process. This implies especially that ${W(t)}_{t \geq 0}$ and $\left{\varepsilon W\left(\varepsilon^{-2} t\right)\right}_{t \geq 0}$ are in law the same process.
Furthermore, let $\mathcal{F}$ be cubic (i.e. $\mathcal{F}(u)=\mathcal{F}(u, u, u)$ is trilinear).

Denote the kernel (or nullspace) of $L$ by $\mathcal{N}$ and the orthogonal projection onto $\mathcal{N}$ by $P_{c}$. Define $P_{s}=I-P_{c}$.
Then we make the following ansatz:
$$u(t)=\varepsilon a\left(\varepsilon^{2} t\right)+\varepsilon^{2} b\left(\varepsilon^{2} t\right)+\varepsilon^{2} \psi(t)+\mathcal{O}\left(\varepsilon^{3}\right)$$
with $a, b \in \mathcal{N}$ and $\psi \in \mathcal{S}:=\mathcal{N}^{\perp}$ the orthogonal complement of $\mathcal{N}$ in $X$.
This ansatz is motivated by the fact that, due to the linear damping of order one in $\mathcal{S}$, the modes in $\mathcal{S}$ are expected to evolve on time scales of order one, whereas the modes in $\mathcal{N}$ are expected to evolve on much slower time scales of order $\varepsilon^{-2}$, as the linear operator is of order $\varepsilon^{2}$. This is mainly due to the fact that we have a well defined spectral gap of order $\mathcal{O}(1)$ between 0 and the first non-zero eigenvalue together with a small linear perturbation of order $\varepsilon^{2}$.

We do not use lower order terms, as we expect that small solutions stay small. Furthermore, using linear and nonlinear stability, it is possible to verify a priori estimates that rigorously verify that the typical scaling of a solution corresponds to the one prescribed by the ansatz (1.4). The statement is called the attractivity result (cf. Section 1.2).

Let us now come back to the formal derivation. Plugging the ansatz (1.4) into (1.2), rescaling to the slow time-scale $T=\varepsilon^{2} t$ and expanding in orders of $\varepsilon$, we obtain by collecting all terms of order $\varepsilon^{3}$ in $\mathcal{N}$
$$\partial_{T} a(T)=A_{c} a(T)+\mathcal{F}{c}(a(T))+\partial{T} \beta(T)$$
Here,
$$\beta(T)=\varepsilon P_{c} Q W\left(\varepsilon^{-2} T\right), \quad T \geq 0$$
is a Wiener process in $\mathcal{N}$ with law independent of $\varepsilon$, due to the scaling properties of the Wiener process. We used
$$A_{c}=P_{c} A \quad \text { and } \quad \mathcal{F}{c}=P{c} \mathcal{F}$$
for short.
This approximating equation in (1.5) is called amplitude equation, as it can by rewritten to an SDE for the amplitudes of an expansion of $a$ with respect to a basis in $\mathcal{N}$. Results like this well known for many examples in the physics and applied mathematics literature (for example [CH93, (4.31), (5.11)]). Moreover, there are numerous variants of this method. However, most of these results are non-rigorous approximations using this type of formal multi-scale analysis.

金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Other Types of Nonlinearities

Cubic nonlinearities are not very special, we can extend the simple idea of the previous section, using scaling and projection, to a lot of different types of nonlinearities. If we look at suitable scalings of the noise and the linear (in)stability we obtain in all cases interesting results. If we do not adapt the scaling, we either loose the linear instability or the noise in the amplitude equation.

Suppose for this section that $\mathcal{F}^{(n)}$ is some multi-linear nonlinearity, which is homogeneous of degree $n \in \mathbb{N}$ with $n \geq 2$ (i.e. for $\alpha>0, \mathcal{F}^{(n)}(\alpha u)=\alpha^{n} \mathcal{F}^{(n)}(u)$ ). Then the noise strength in the SPDE $(1.2)$ should be changed to $\sigma^{2}=\varepsilon^{(n+1) /(n-1)}$ instead of $\varepsilon^{2}$. Thus the equation reads in the interesting scaling
$$\partial_{t} u=L u+\varepsilon^{2} A u+\mathcal{F}^{(n)}(u)+\varepsilon^{(n+1) /(n-1)} \xi .$$

Now with the ansatz
$$u(t)=\varepsilon^{2 /(n-1)} a\left(\varepsilon^{2} t\right)+\varepsilon^{(n+1) /(n-1)} \psi(t)+\mathcal{O}\left(\varepsilon^{2 n /(n-1)}\right)$$
and a similar formal calculation as in the previous section, we derive the same type of amplitude equation. First collecting all terms of order $\varepsilon^{2 n /(n-1)}$ in $\mathcal{N}$ yields
$$\partial_{T} a=P_{c} A a+P_{c} \mathcal{F}^{(n)}(a)+\partial_{T} \beta$$
which now contains a nonlinearity which is homogeneous of degree $n$. The second order correction is exactly the same (cf. $(1.6))$ as in the cubic case, but now it contains all terms in $\mathcal{S}$ of order $\varepsilon^{(n+1) /(n-1)}$.

We will not focus on rigorous results for this type of equations, as they are very similar to the cubic case. After minor modifications one can easily transfer all results to the general case.

金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Formal Derivation of Amplitude Equations

∂吨在=大号在+e2一种在+F(在)+e2X,

• 大号是一个对称的非正微分算子( 例如 1+∂X2)2) 具有非零有限维内核（或零空间）。
• e2一种在是一个小的（线性）确定性扰动，
• F是一些非线性，例如像一个稳定的立方−在3.
• X=X(吨,X)是空间和时间上的高斯噪声
我们稍后给出噪声的例子，它在时间上被认为是白色的，在空间上可以是白色或彩色的。更准确地说，假设X是一个广义高斯过程，使得对于均值和相关
和X(吨,X)=0 和 和X(吨,X)X(s,是)=d(吨−s)q(X−是)
对于一些合适的空间相关函数（或分布）q. 如果q是 Delta 分布d, 那么我们调用X时空白噪声。在这种情况下X=∂吨在是圆柱维纳过程的广义导数在(吨)吨≥0在合适的希尔伯特空间中。

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

∂吨在=−(1+Δ)2在+e2ν在−在3+e2∂吨问在.

$$\partial_{T} a(T)=A_{c} a(T)+\mathcal{F} {c}(a(T))+\partial {T} \beta(T) H和r和, \beta(T)=\varepsilon P_{c} QW\left(\varepsilon^{-2} T\right), \quad T \geq 0 一世s一种在一世和n和rpr这C和ss一世nñ在一世吨Hl一种在一世nd和p和nd和n吨这Fe,d在和吨这吨H和sC一种l一世nGpr这p和r吨一世和s这F吨H和在一世和n和rpr这C和ss.在和在s和d A_{c}=P_{c} A \quad \text { 和 } \quad \mathcal{F} {c}=P {c} \mathcal{F}$$

（1.5）中的这个近似方程称为幅度方程，因为它可以重写为 SDE 的展开的幅度一种关于基础ñ. 这样的结果在物理学和应用数学文献中的许多例子中众所周知（例如 [CH93, (4.31), (5.11)]）。此外，这种方法有许多变体。然而，这些结果中的大多数是使用这种形式的多尺度分析的非严格近似。

金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Other Types of Nonlinearities

∂吨在=大号在+e2一种在+F(n)(在)+e(n+1)/(n−1)X.

∂吨一种=磷C一种一种+磷CF(n)(一种)+∂吨b

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