### 金融代写|随机偏微分方程代写Stochastic partialAssumptions

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

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

Let us summarise all assumptions necessary for our results. We do not focus on the highest possible level of generality, but stick to some simpler setting which cover all our examples. First consider the linear operator $L$.

Assumption 2.5 Let $X$ be a separable Hilbert space and $\Delta$ (subject to some boundary conditions on a bounded domain) be a self-adjoint version of the Laplacian on $X$. Suppose $L=P(-\Delta)$ for some function $P$ such that $L$ is non-positive. Furthermore, let the kernel $\mathcal{N}=\operatorname{ker} L$ of $L$ be non-empty and finite dimensional. Finally, suppose $P(k) \rightarrow-\infty$ as $k \rightarrow \infty$.

This assumption is a stronger than the one in Section 2.2. It is mainly used for convenience of presentation, and covers all examples presented. Furthermore, it is just a special case of Assumption 2.1, and in the following we can use all the implications of this assumption. We use the notation $P_{c}$ and $P_{s}$, which are in this case just the standard orthogonal projections. Additionally, recall the splitting $X=\mathcal{N} \oplus \mathcal{S}$ with $\mathcal{S}=P_{s} X$ and the spaces $X^{\alpha}$ from Section 2.2. Recall furthermore the bounds (2.4), (2.5), and (2.6) for the analytic semigroup $\mathrm{e}^{t L}$ generated by $L$.
For the nonlinearities, we make two assumptions. The first one, is much weaker than Assumption 2.2, as we are aiming only for local results in that case. Especially, we can get rid of the strong nonlinear dissipativity. The second assumption is similar to Assumption $2.2$ and involves strong nonlinear stability and dissipativity conditions in $\mathcal{N}$.

Assumption 2.6 The function $\mathcal{F}$ is locally Lipschitz from $X$ to $X^{-\alpha}$ for some $\alpha \in[0,1)$. This means that for all $R>0$ there is a $C>0$ such that
$$\left|\mathcal{F}\left(v_{1}\right)-\mathcal{F}\left(v_{2}\right)\right|_{-\alpha} \leq C\left|v_{1}-v_{2}\right| \quad \text { for all } v_{i} \text { with }\left|v_{i}\right| \leq R$$
Assume we can split $\mathcal{F}(x)=f(x)+g(x)$, where $f: X \times X \times X \rightarrow X^{-\alpha}$ is continuous, trilinear, and symmetric. The function $g$ is of higher order, which means $|g(x)|_{-\alpha} \leq C|x|^{4}$ provided $|x| \leq 1 .$

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Amplitude Equations Main Results

Theorem 2.7 (Attractivity-local) Under Assumptions $2.5,2.6$, and $2.8$ fix some small constant $\kappa>0$. Then there are constants $c_{i}>0$ and a time $t_{e}=$ $\mathcal{O}\left(\ln \left(\varepsilon^{-1}\right)\right)$ such that for all mild solutions $u$ of $(2.86)$ we can write
$$u\left(t_{e}\right)=\varepsilon a_{e}+\varepsilon^{2} R_{e} \quad \text { with } \quad a_{\varepsilon} \in \mathcal{N} \text { and } R_{e} \in \mathcal{S}$$
where
$$\mathbb{P}\left(\left|a_{e}\right| \leq \delta,|R e| \leq \varepsilon^{-\kappa}\right) \geq \mathbb{P}\left(|u(0)| \leq c_{3} \delta \varepsilon\right)-c_{1} e^{-c_{2} e^{-2 \kappa}}$$
for all $\delta>1$ and $\varepsilon \in(0,1)$.
This result states in a weak form that $u(0)=\mathcal{O}(\varepsilon)$ with high probability implies $P_{c} u\left(t_{\varepsilon}\right)=\mathcal{O}(\varepsilon)$ and $P_{s} u\left(t_{\varepsilon}\right)=\mathcal{O}\left(\varepsilon^{2}\right)$ with high probability, too. Note that we do not bound any moments of the solution $u$.

We do not give a detailed proof of this result, as it is a straightforward modification of Theorem $3.3$ of [Blö03]. It relies on the fact that small solutions of order $\mathcal{O}(\varepsilon)$ are on small time-scales given by the linearised picture, which is dominated by the semigroup estimates $(2.5)$ and $(2.6)$. Thus modes in $P_{s} X$ decay exponentially fast on a time-scale of order $\mathcal{O}(1)$.
Using strong nonlinear stability, we can prove much more:
Theorem 2.8 (Attractivity-global) Let Assumptions 2.5, 2.6, and 2.8 be satisfied. Then for all times $T_{e}=T_{0} \varepsilon^{-2}>0$ and for all $p \geq 1$ there are constants $C_{p}>0$ explicitly depending on $p$ such that
$$\mathbb{E}\left|u\left(t+T_{e}\right)\right|^{p} \leq C_{p} \varepsilon^{p} \quad \text { and } \quad \mathbb{E}\left|P_{s} u\left(t+T_{e}\right)\right|^{p} \leq C_{p} \varepsilon^{2 p}$$
for all $t \geq 0$, all $X$-valued mild solutions $u$ of equation (1.2) independent of the initial condition $u(0)$, and for all $\varepsilon \in(0,1)$.

Furthermore, if we already assume that $\mathbb{E}|u(0)|{ }^{p} \leq C_{p} \varepsilon^{p}$ for a constant $C_{p}>0$, then there is a time $t_{\varepsilon}=\mathcal{O}\left(\ln \left(\varepsilon^{-1}\right)\right)$ and a constant $C>0$ such that
$$\mathbb{E}|u(t)|^{p} \leq C \varepsilon^{p} \quad \text { and } \quad \mathbb{E}\left|P_{s} u\left(t+t_{e}\right)\right|^{p} \leq C \varepsilon^{2 p}$$
for all $t \geq 0$, all $X$-valued mild solutions $u$, and for all $\varepsilon \in(0,1)$.
The proof is given by a priori estimates. This was not directly proved in [BH04], but under our somewhat stronger assumptions this is similar to Lemma $4.3$ of [BH04]. It relies on a priori estimates for $v_{\delta_{\varepsilon}}=u-\varepsilon^{2} W_{L-\delta_{c}}$ with $\delta_{e}=\mathcal{O}\left(\varepsilon^{2}\right)$, which fulfils a random PDE similar to $(2.87)$. The main technical advantage is that the linear semigroup generated by $L-\delta_{e}$ is exponentially stable.
2.5.3.2 Approximation
For a solution $a$ of $(1.5)$ and $\psi$ of $(1.6)$ we define the approximations $\varepsilon w_{k}$ of order $k$ by
$$\varepsilon w_{1}(t):=\varepsilon a\left(\varepsilon^{2} t\right) \text { and } \varepsilon w_{2}(t):=\varepsilon a\left(\varepsilon^{2} t\right)+\varepsilon^{2} \psi(t)$$

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

In this section we review the results of [Blö05a]. Consider an SPDE of the following type.
$$\partial_{t} u=L u+\varepsilon^{2} A u+B(u, u)+\varepsilon^{2} \xi$$
where $L$ as in Assumption 2.1. The linear operator $A$ and the bilinear operator $B$ are as in Assumption 2.4, and the noise is the generalised derivative of some Wiener process (cf. Assumption 2.9).

In [Blö05a] we used fractional noise. This was motivated by the fact that the proofs rely on fractional integration by parts formulas, and explicit path-wise estimates. Here we state for simplicity only the version for Gaussian noise that is white in time. Note that due to the method of proof, we need the noise to be trace-class, as we need bounds for the Wiener process $W(t)$ in the space $X$. This obviously rules out space-time white noise.

Let us furthermore point out that the Hilbert space setting is not necessary in this approach, as we purely rely on local results, using cut-off techniques, and we do not use a priori estimates. It is also necessary to deal with non self-adjoint operators, as the linear part in the Rayleigh-Bénard system is not self-adjoint (cf. Section $6.1$ of [Blö05a] for a detailed discussion.
For the stochastic perturbation $\xi$ let the following assumption be true.
Assumption 2.9 (Noise) Suppose that the noise process $\xi$ is the generalised derivative of some Wiener process ${Q W(t)}_{t \geq 0}$ on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$, where $W$ is the standard cylindrical Wiener process.
Assume that the stochastic convolution
$$W_{L}(t)=\int_{0}^{t} \mathrm{e}^{(t-\tau) L} d Q W(\tau)$$
is a well defined stochastic process with continuous paths in $X$. We suppose that the noise (or $W)$ is of trace-class, i.e. $\operatorname{tr}\left(Q^{2}\right)=\mathbb{E}|Q W(t)|^{2}<\infty$.

This assumption is stronger than Assumption 2.8. Especially, $W$ being traceclass is a serious restriction, as this already implies that $W$ has continuous paths in $X$. We briefly sketched after Remark $2.7$ the connection between the spatial correlation function $q$ of the noise $\xi$ and the operator $Q$ belonging to $W$. The condition of $W$ being trace-class is essentially a regularity condition on $q$. See for example [Blö05b]. Any decay condition for the eigenvalues of $Q$ immediately transfers to a decay condition of the Fourier coefficients of $q$.
To give a meaning to $(2.93)$ we consider as usual mild solutions.

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

|F(在1)−F(在2)|−一种≤C|在1−在2| 对全部 在一世 和 |在一世|≤R

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Amplitude Equations Main Results

2.5.3.2 近似

e在1(吨):=e一种(e2吨) 和 e在2(吨):=e一种(e2吨)+e2ψ(吨)

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

∂吨在=大号在+e2一种在+乙(在,在)+e2X

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