### 金融代写|随机偏微分方程代写Stochastic partialResults for 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|Results for Quadratic Nonlinearities

This section states rigorous results for the approximation via amplitude equations for quadratic nonlinearities. We focus only on the interesting case, where $P_{c} B(a, a)=0$, which was discussed for additive noise on a formal level in Section 1.1.3. The case with $P_{c} B(a, a) \neq 0$ is similar to the cubic case. The formal result for our case is completely analogous to the one stated in Section 1.1.2, we summarise details below. Nevertheless, in this case in general we cannot bound moments of solutions. We have to use cut-off techniques in order to use moments.

Here we present a somewhat simpler model with multiplicative noise, in order to simplify the presentation. We review the results of [Blö05a] for additive noise in

Section 2.6. In [Blö05a] also fractional (i.e. smoother) additive noise was used, but we do not focus on that.
Consider
$$\partial_{t} u=L u+\varepsilon^{2} A u+B(u, u)+\varepsilon u \dot{\beta},$$
with $L$ and $A$ as in Assumption $2.1$ and $2.2$, and $B$ some bilinear mapping defined later on in Assumption 2.4.

Let us recall the formal derivation of the amplitude equation, which is similar to Section 1.1.3. Plugging the ansatz
$$u(t)=\varepsilon a\left(\varepsilon^{2} t\right)+\varepsilon^{2} \psi_{o}\left(\varepsilon^{2} t\right)$$
with $a \in \mathcal{N}$ and $\psi_{o} \in P_{s} X$ into (2.40), we derive in lowest order of $\varepsilon>0$
$\mathcal{O}\left(\varepsilon^{2}\right)$ in $\mathcal{N}: \quad 0=B_{c}(a, a)$,
$\mathcal{O}\left(\varepsilon^{3}\right)$ in $\mathcal{N}: \quad \partial_{T} a=A_{c} a+2 B_{c}\left(a, \psi_{o}\right)+a \partial_{T} \tilde{\beta}$,
$\mathcal{O}\left(\varepsilon^{2}\right)$ in $P_{s} X: \quad 0=L \psi_{o}+B_{s}(a, a) .$
Note that $\tilde{\beta}(T)=\varepsilon \beta\left(T \varepsilon^{-2}\right)$ is again a rescaled Brownian motion. From (2.41) we see that $B_{c}(a, a)=0\left(B_{c}:=P_{c} B\right.$, as usual $)$ is necessary for the approach presented. Finally, projecting (2.43) to $P_{s} X$ and solving for $\psi_{o}$ yields
$$\partial_{T} a=A_{c} a-2 B_{c}\left(a, L_{s}^{-1} B_{s}(a, a)\right)+a \partial_{T} \tilde{\beta}$$
or in integrated form
$$a(T)=a(0)+\int_{0}^{T}\leftA_{c} a-2 B_{c}\left(a, L_{s}^{-1} B_{s}(a, a)\right)\right d \tau+\int_{0}^{T} a(\tau) d \tilde{\beta}(\tau)$$
where we consider as before Itô-differentials. Nevertheless, as discussed before in Section 2.1, we could also consider Stratonovič-differentials everywhere, and still obtain the same result. An interesting feature of (2.45) is that the amplitude equation involves a cubic nonlinearity. Therefore, we can expect nonlinear stability of the amplitude equation, which is in general not present for the SPDE.

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

We use a cut-off technique, as in general we cannot control moments of solutions. There are some special cases like for instance one-dimensional Burgers, surface growth, or Kuramoto-Sivashinsky equation (see [BGR02; DPDT94; DE01]), where we actually can derive bounds for moments. But for our results it is enough to cut off the nonlinearity for large solutions, in order to keep it small for solutions that get too large.

This technique is well known for SDEs with blow-ups. See for example [McK69]. For a detailed discussion see Section $6.3$ of [HT94]. The idea is always to cut off the nonlinearities, in order to derive bounds for moments and to compute probabilities. But solutions of the modified equation with cut-off and the original equation coincide, as long as both are small. Note that for the local attractivity result we are anyway only interested in solutions that are small. To be more precise we look at solutions of order $\mathcal{O}(\varepsilon)$.

The main result is a local attractivity result for solutions of order $\mathcal{O}(\varepsilon)$. It shows that if $u(0)$ is of order $\mathcal{O}(\varepsilon)$, then at some time $t_{\varepsilon}=\mathcal{O}\left(\ln \left(\varepsilon^{-1}\right)\right)$ the probability is almost 1 that $u\left(t_{\varepsilon}\right)$ is still of order $\mathcal{O}(\varepsilon)$, but $P_{s} u\left(t_{e}\right)$ decreased to order $\mathcal{O}\left(\varepsilon^{2}\right)$.
Theorem 2.4 (Attractivity) Let Assumptions 2.1, 2.3, and 2.4 be true.
For all small $\kappa>0$, all $\delta>0$ and $p>0$ there are constants $C>0, \delta_{1}, \delta_{2}>0$ such that for $t_{e}=\frac{2}{\omega} \ln \left(\varepsilon^{-1}\right)$ and all mild solutions in the sense of Definition $2.5$
$$\mathbb{P}\left(\left|u\left(t_{\varepsilon}\right)\right| \leq \delta_{1} \varepsilon,\left|P_{s} u\left(t_{\varepsilon}\right)\right| \leq \delta_{2} \varepsilon^{2}\right) \geq \mathbb{P}(|u(0)| \leq \delta \varepsilon)-C \varepsilon^{p}$$
for all $\varepsilon \in(0,1)$.
The proof relies on the linear stability of (2.40) and cut-off techniques. We postpone the proof to Section $2.4 .4$ as it is not difficult but technical. Let us first discuss the results for the residual and the approximation.

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

For a solution of the amplitude equation $(2.45)$ and some $\psi(0)$, we consider the approximation $\varepsilon w$ given by $(2.47)$. The residual of $\varepsilon w$ is as usual defined as
\begin{aligned} \operatorname{Res}(\varepsilon w)(t)=&-\varepsilon w(t)+\varepsilon \mathrm{e}^{t L} w(0)+\varepsilon^{2} \int_{0}^{t} \mathrm{e}^{(t-\tau) L} w(\tau) d \beta(\tau) \ &+\int_{0}^{t} \mathrm{e}^{(t-\tau) L}\left\varepsilon^{3} A w+\varepsilon^{2} B(w)\right d \tau \end{aligned}
Theorem 2.5 (Residual) Let Assumptions 2.1, 2.3, and 2.4 be true.
For $p>4, \delta>0, T_{0}>0$ there is a constant $C>0$ such that for all approximations defined by (2.46) and (2.47), where a is a solution of (2.45), with $\mathbb{E}|a(0)|^{4 p} \leq \delta$ and $\mathbb{E}|\psi(0)|^{2 p} \leq \delta$ we have
$$\mathbb{E}\left(\sup {t \in\left[0, T{0} \varepsilon^{-2}\right]}\left|P_{c} \operatorname{Res}(\varepsilon w)(t)\right|^{p}\right) \leq C \varepsilon^{2 p}$$
and
$$\mathbb{E}\left(\sup {t \in\left[0, T{0} \epsilon^{-2}\right]}\left|P_{s} \operatorname{Res}(\varepsilon w)(t)\right|^{p}\right) \leq C \varepsilon^{3 p-2}$$
Furthermore $P_{c} \operatorname{Res}(\varepsilon w)$ is differentiable with
$$\partial_{t} P_{c} \operatorname{Res}(\varepsilon w)(t)=\varepsilon^{4}\left[A_{c} \psi+B_{c}(\psi)\right]\left(\varepsilon^{2} t\right)$$
The proof below is straightforward using the key Lemma 2.2. This lemma is a purely technical estimate, and we postpone the proof to Section 2.4.4.

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

∂吨在=大号在+e2一种在+乙(在,在)+e在b˙,

∂吨一种=一种C一种−2乙C(一种,大号s−1乙s(一种,一种))+一种∂吨b~

$$a(T)=a(0)+\int_{0}^{T}\left A_{c} a-2 B_{c}\left(a, L_{s}^{- 1} B_{s}(a, a)\right)\right d \tau+\int_{0}^{T} a(\tau) d \tilde{\beta}(\tau)$$

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

\begin{aligned} \operatorname{Res}(\varepsilon w)(t)=&-\varepsilon w(t)+\varepsilon \mathrm{e}^{t L} w(0 )+\varepsilon^{2} \int_{0}^{t} \mathrm{e}^{(t-\tau) L} w(\tau) d \beta(\tau) \ &+\int_{ 0}^{t} \mathrm{e}^{(t-\tau) L}\left \varepsilon^{3} A w+\varepsilon^{2} B(w)\right d \tau \end{对齐} 吨H和这r和米2.5(R和s一世d在一种l)大号和吨一种ss在米p吨一世这ns2.1,2.3,一种nd2.4b和吨r在和.F这rp>4,d>0,吨0>0吨H和r和一世s一种C这ns吨一种n吨C>0s在CH吨H一种吨F这r一种ll一种ppr这X一世米一种吨一世这nsd和F一世n和db是(2.46)一种nd(2.47),在H和r和一种一世s一种s这l在吨一世这n这F(2.45),在一世吨H和|一种(0)|4p≤d一种nd和|ψ(0)|2p≤d在和H一种在和 \mathbb{E}\left(\sup {t \in\left[0, T{0} \varepsilon^{-2}\right]}\left|P_{c} \operatorname{Res}(\varepsilon w )(t)\right|^{p}\right) \leq C \varepsilon^{2 p} 一种nd \mathbb{E}\left(\sup {t \in\left[0, T{0} \epsilon^{-2}\right]}\left|P_{s} \operatorname{Res}(\varepsilon w )(t)\right|^{p}\right) \leq C \varepsilon^{3 p-2} F在r吨H和r米这r和磷C水库⁡(e在)一世sd一世FF和r和n吨一世一种bl和在一世吨H \partial_{t} P_{c} \operatorname{Res}(\varepsilon w)(t)=\varepsilon^{4}\left[A_{c} \psi+B_{c}(\psi)\right] \left(\varepsilon^{2} t\right)

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