### 金融代写|随机偏微分方程代写Stochastic partial differentialAmplitude Equations on Bounded Domains

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

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

Let us first motivate, why we are interested in multiplicative noise. It appears naturally in models, where one considers noisy control parameters. Consider as an example some deterministic PDE of the following type
where $L$ is some linear differential operator and $\mathcal{F}$ is some nonlinearity, for instance $-u^{3}$. Suppose that the equation undergoes a change of stability (or bifurcation) when $\mu=0$.

The question is, whether we can see the influence of small noise in the bifurcation parameter $\mu$ in the case where $\mu$ is near or at the bifurcation. This is an important question in many experiments, as $\mu$ models experimental quantities like, for instance, temperature, which are naturally subject to small (random) perturbations.

We consider in (2.1) a simplified PDE model, where the perturbation of the parameter has no spatial dependence and is homogeneous in space. This kind of equation was recently studied in more detail, for instance, by [CLR00; CLR01; Rob02] where they determined the dimension and structure of a random attractor for a stochastic Ginzburg-Landau equation. On the other hand, even the stability of linear equations (i.e. $\mathcal{F} \equiv 0$ ) was only studied recently in [CR04] or [Kwi02] following the celebrated work of [ACW83].

Let us come back to (2.1). Assume that the control parameter $\mu \in \mathbb{R}$ is perturbed by white noise and suppose the strength of the fluctuations $\varepsilon>0$ is small. A typical model is a Gaussian noise $\mu$ with some mean and covariance functional
$$\mathbb{E} \mu(t)=\mu_{\varepsilon} \in \mathbb{R}, \quad \mathbb{E}\left(\mu(t)-\mu_{\varepsilon}\right)\left(\mu(s)-\mu_{\varepsilon}\right)=\varepsilon^{2} \delta(t-s) .$$
Thus we can write $\mu=\mu_{\varepsilon}+\varepsilon \xi$, where $\xi=\partial_{t} \beta$ is the generalised derivative of a real valued Brownian motion $\beta={\beta(t)}_{t \geq 0}$.
Hence, we can rewrite (2.1) as a stochastic PDE
$$\partial_{t} u=L u+\mu_{\varepsilon} u+\mathcal{F}(u)+\varepsilon u \partial_{t} \beta$$

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Assumptions and Results The Cubic Case

This section summarises for the cubic case all assumptions necessary and states the main results. In this chapter we treat two sets of different assumptions. On one hand this section treats nonlinear stable equations involving cubic terms, where we can use standard a priori estimates to obtain bounds on moments of solutions. On the other hand we consider in Section $2.4$ quadratic nonlinearities, which in general do not allow to bound moments of solutions. Especially, if we cannot rule out the possibility of a blow-up of solutions in finite time, which is the case in many examples. One is the 2D Kuramoto-Sivashinsky equation, for instance. In this case we obtain local result by using cut-off techniques.

Consider the following SPDE in some Hilbert space $X$ with scalar product $\langle\cdot,\rangle$, and norm $|\cdot|$. We could also consider Banach spaces here, but the Hilbert space setting simplifies the notation and the a priori estimates on solutions.
$$d u=\left[L u+\varepsilon^{2} A u+\mathcal{F}(u)\right] d t+\varepsilon u d \beta$$
The precise setting is given below in Assumptions $2.1$ for $L, 2.2$ for $A$ and $\mathcal{F}$, and $2.3$ for the Itô-differential.

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

We establish two results. The first one in Theorem $2.1$ is a very strong result. It relies on the nonlinear stability of the equation and establishes bounds on $\mathbb{E}|u(t)|^{p}$ for large $t$ completely independent of the initial condition $u(0)$. The second result is somewhat weaker. It relies on the existence of bounds on $\mathbb{E}|u(0)|^{p}$, and it establishes bounds on $\mathbb{E}\left|P_{s} u(t)\right|^{p}$ for moderately large $t$. This relies mainly on the linearised picture and a spectral gap of the linearised operator.

For the attractivity our main goal is to verify that there is a time $t_{\varepsilon}>0$ such that
$$u\left(t_{e}\right)=\varepsilon a_{e}+\varepsilon^{3} \psi_{e},$$
where $a_{\varepsilon} \in \mathcal{N}$ and $\psi_{e} \in P_{s} X$ are both of order $\mathcal{O}(1)$.
Theorem 2.1 (Attractivity) Let Assumptions 2.1, 2.2, and $2.3$ be true and let $u$ be a strong solution of (2.3) in $X$.
Then for all $p>0$ and $t_{0}>0$ there is a constant $C>0$ such that
$$\sup {t \geq t{0} e^{-2}} \mathbb{E}|u(t)|^{p} \leq C \varepsilon^{p}$$
for all sufficiently small $\varepsilon>0$ and all strong solutions $u$ of (2.3) in $X$ independent of the initial condition. Especially, $\tau_{e}=\infty$ almost surely for the maximal time of existence of $u$.

Furthermore, for $q \geq 2, \delta>0$, and $p \in[2, q]$ there is some constant $C>0$ such

that $\mathbb{E}|u(0)|^{q} \leq \delta \varepsilon^{q}$ for all $\varepsilon \in(0,1)$ implies
$$\sup {t \geq 0} \mathbb{E}|u(t)|^{p} \leq C \varepsilon^{p} \quad \text { for all sufficiently small } \varepsilon>0 \text {. }$$ Additionally, for $t{e}=\frac{2}{\omega} \ln \left(\varepsilon^{-1}\right)$ and all $p \in[4, q / 3]$ there is a constant $C>0$ such that
$$\sup {t \geq t{\varepsilon}} \mathbb{E}\left|P_{s} u(t)\right|^{p} \leq C \varepsilon^{3 p} \quad \text { for all sufficiently small } \varepsilon>0 .$$
The proof is straightforward. But, as it is quite technical, we postpone it to Section 2.3. The main tools are standard a priori type estimates using Itô’s formula and Burkholder’s inequality.

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

∂吨在=大号在+μe在+F(在)+e在∂吨b

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Assumptions and Results The Cubic Case

d在=[大号在+e2一种在+F(在)]d吨+e在db

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