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## 统计代写|离散时间鞅理论代写martingale代考|The Corrector Field

In this subsection, we introduce the corrector field and establish an energy identity. Recall the resolvent equation (3.16) for the local drift $V$. Multiplying both sides of the resolvent equation by $f_{j, \lambda}$ and integrating over $\Omega$, in view of the explicit formula (3.12) for the $\mathscr{H}{1}$ norm and of relation (3.21), $$\lambda\left|f{j, \lambda}\right|_{\mathbb{Q}}^{2}+\frac{1}{2} \sum_{z \in \Lambda}\left\langle p_{z} D_{z} f_{j, \lambda}, D_{z} f_{j, \lambda}\right\rangle_{\mathbb{Q}}=\sum_{z \in \Lambda}\left\langle p_{z} H_{j, z}, D_{z} f_{j, \lambda}\right\rangle_{\mathbb{Q}}$$
for $1 \leq j \leq d$. Since $H_{j, z}$ belongs to $L^{2}(\mathbb{Q})$ and since $p_{z}$ is strictly elliptic,
$$\sup {0<\lambda \leq 1} \lambda\left|f{j, \lambda}\right|_{\mathbb{Q}}^{2}<\infty \text { and } \sup {0<\lambda \leq 1}\left|D{z} f_{j, \lambda}\right|_{\mathbb{Q}}^{2}<\infty$$
for all $z$ in $\Lambda, 1 \leq j \leq d$
Proposition 3.7 Suppose that the hypotheses of Theorem $3.6$ hold and that $F_{j}$ is the weak limit in $\mathscr{H}{1}$ of $\left{f{j, \lambda_{n}}: n \geq 1\right}$, where $\lambda_{n} \downarrow 0$, as $n \uparrow \infty$. Then, for $1 \leq j \leq d$,
$$\frac{1}{2} \sum_{z \in \Lambda}\left\langle p_{z} D_{z} F_{j}, D_{z} F_{j}\right\rangle_{\mathbb{Q}}=\sum_{z \in \Lambda}\left\langle p_{z} H_{j, z}, D_{z} F_{j}\right\rangle_{\mathbb{Q}} .$$
This proposition is proved in three steps. In this subsection, we define the corrector field and we obtain two estimates on its asymptotic behavior. In the next subsection, we derive an elliptic difference equation satisfied by the corrector field from which we deduce, in the last subsection, the energy estimate stated above.
Fix a sequence $\left{\lambda_{n}: n \geq 1\right}$ which vanishes as $n \uparrow \infty$ and such that $f_{j, \lambda}$ converges weakly to $F_{j}$ in $\mathscr{H}{1}$ for $1 \leq j \leq d$. Since $\left{D{z} f_{j, \lambda}: 0<\lambda<1\right}$ is a bounded sequence in $L^{2}(\mathbb{Q})$, taking a further subsequence if necessary, we may assume that $D_{z} f_{j, \lambda_{n}}$ converges weakly in $L^{2}(\mathbb{Q})$ for all $z$ in $\Lambda, 1 \leq j \leq d$. Let
$$f_{j}(z)-\lim {n \rightarrow \infty} D{z} f_{j, \lambda_{n}} .$$
Clearly, $\mathscr{D} F_{j}=\left{f_{j}(z): z \in \Lambda\right}$ in $L^{2}(\mathbb{Q})$.

## 统计代写|离散时间鞅理论代写martingale代考|An Elliptic Equation for the Corrector Field

In this subsection we demonstrate that the corrector field satisfies in a weak sense a linear elliptic difference equation $\mathbb{Q}$-a.s. Let $C_{0}\left(\mathbb{Z}^{d}\right)$ be the space of all compactly supported functions on $\mathbb{Z}^{d}$. Recall the definition of the random variables $\left{H_{j, z}: z \in\right.$ $\Lambda}, 1 \leq j \leq d$, introduced in (3.21) and recall that $q_{z}=p_{z}-p_{-z} \circ \tau_{z}$

Proposition 3.11 Let $\left{f_{j}(x): x \in \mathbb{Z}^{d}\right}, 1 \leq j \leq d$, be a corrector field. For each $1 \leq j \leq d$ and for every $x$ in $\mathbb{Z}^{d}, \mathbb{Q}$-a.s.,
\begin{aligned} &\frac{1}{2} \sum_{z \in \Lambda} \partial_{z}^{}\left{p(x, x+z) \partial_{z} f_{j}(x)\right}-\frac{1}{2} \sum_{z \in \Lambda^{s}} q(x, x+z) \partial_{z} f_{j}(x) \ &\quad=\sum_{z \in \Lambda} \partial_{z}^{}\left{p(x, x+z) H_{j, z}(x)\right} \end{aligned}

where $q(x, x+z):=q_{z} \circ \tau_{x}, \Lambda^{s}=\left{z \in \mathbb{Z}^{d}: z\right.$ or $\left.-z \in \Lambda\right}$ and $\partial_{z}^{}$ represents the adjoint of $\partial_{z}:\left(\partial_{z}^{} g\right)(x)=[g(x-z)-g(x)]$.

Proof Fix a corrector field $\left{f_{j}(x): x \in \mathbb{Z}^{d}\right}, 1 \leq j \leq d$, and consider a sequence $\left{\lambda_{n}: n \geq 1\right}$, vanishing as $n \uparrow \infty$, for which (3.24) holds weakly in $L^{2}(\mathbb{Q})$ for all $x \in \mathbb{Z}^{d}$. Multiply both sides of the resolvent equation (3.3) by $g \in L^{2}(\mathbb{Q})$ and integrate with respect to $\mathbb{Q}$. By $(3.21)$, the right-hand side is equal to $\sum_{z \in \Lambda}\left\langle p_{z} H_{j, z}, D_{z} g\right\rangle_{\mathbb{Q}}$. On the left-hand side, $\left\langle\lambda_{n} f_{\lambda_{n}, j}, g\right\rangle_{\mathbb{Q}}$ vanishes as $n \uparrow \infty$ because $\lambda_{n} f_{\lambda_{n}, j}$ vanishes in $L^{2}(\mathbb{Q})$. Rewriting the generator $L$ as $S+A$, and recalling the explicit formulas $(3.10),(3.11)$ for $S$ and $A$ as well as the definition of $f_{j}(z)$, we obtain that
$$2 \sum_{z \in \Lambda}^{1}\left\langle p_{z} f_{j}(z), D_{z} g\right\rangle_{\mathbb{Q}}-\sum_{2 \in \Lambda^{s}}^{1}\left\langle q_{z} f_{j}(z), g\right\rangle_{\mathbb{Q}}=\sum_{z \in \Lambda}\left\langle p_{z} H_{j, z}, D_{z} g\right\rangle_{\mathbb{Q}} .$$

## 统计代写|离散时间鞅理论代写martingale代考|The Energy Identity

In this subsection, we prove the energy identity stated in Proposition 3.7. Let $\varphi$ : $\mathbb{R}^{d} \rightarrow[0, \infty)$ be any smooth, non-negative function, supported in the unit cube $(-1,1)^{d}$ and satisfying $\int_{\mathbb{R}^{d}} \varphi(x) d x=1$. For $\varepsilon>0$, let $\varphi_{\varepsilon}(x)=\varepsilon^{d} \varphi(\varepsilon x)$.

Replace in (3.31) $h, \Phi$ by $f_{j}, \varphi_{\varepsilon}$, respectively. We claim that the first term on the left-hand side of (3.31) converges in probability, as $\varepsilon \downarrow 0$, to
$$\frac{1}{2} \sum_{z \in \Lambda}\left\langle p_{z} D_{z} F_{j}, D_{z} F_{j}\right\rangle_{\mathbb{Q}} .$$
In this subsection, convergence in probability refers to the measure $\mathbb{Q}$.
To prove (3.32), note that the first term on the left-hand side of $(3.31)$, with $f_{j}$, $\varphi_{\varepsilon}$ in place of $h, \Phi$, is equal to
$$\frac{1}{2} \sum_{z \in \Lambda} \sum_{x \in \mathbb{Z}^{d}} p(x, x+z) \partial_{z} f_{j}(x) \partial_{z}\left[f_{j}(x) \varphi_{\varepsilon}(x)\right]$$
In view of the formula for the discrete gradient of a product
$$\partial_{z}f g=f(x+z)\left(\partial_{z} g\right)(x)+g(x)\left(\partial_{z} f\right)(x),$$
the previous expression is equal to
\begin{aligned} &\frac{1}{2} \sum_{z \in \Lambda} \sum_{x \in \mathbb{Z}^{d}} \dot{\varphi}{\varepsilon}(x+z) p(x, x+z) \partial{z} f_{j}(x)^{2} \ &\quad+\frac{1}{2} \sum_{z \in \Lambda} \sum_{x \in \mathbb{Z}^{d}} \partial_{z} \varphi_{\varepsilon}(x) p(x, x+z) \partial_{z} f_{j}(x) f_{j}(x) \end{aligned}

## 统计代写|离散时间鞅理论代写martingale代考|The Corrector Field

$$\lambda|f j, \lambda|{\mathbb{Q}}^{2}+\frac{1}{2} \sum{z \in \Lambda}\left\langle p_{z} D_{z} f_{j, \lambda}, D_{z} f_{j, \lambda}\right\rangle_{\mathbb{Q}}=\sum_{z \in \Lambda}\left\langle p_{z} H_{j, z}, D_{z} f_{j, \lambda}\right\rangle_{\mathbb{Q}}$$

$$\sup 0<\lambda \leq 1 \lambda|f j, \lambda|{\mathbb{Q}}^{2}<\infty \text { and } \sup 0<\lambda \leq 1\left|D z f{j, \lambda}\right|{\mathbb{Q}}^{2}<\infty$$ 对所有人 $z$ 在 $\Lambda, 1 \leq j \leq d$ 命题 $3.7$ 假设定理的假设 $3.6$ 持有，那 $F{j}$ 是弱极限 $\mathscr{H} 1$ 1的\1eft 的分隔符缺失或无法识别

$$\frac{1}{2} \sum_{z \in \Lambda}\left\langle p_{z} D_{z} F_{j}, D_{z} F_{j}\right\rangle_{\mathbb{Q}}=\sum_{z \in \Lambda}\left\langle p_{z} H_{j, z}, D_{z} F_{j}\right\rangle_{\mathbb{Q}}$$

$$f_{j}(z)-\lim n \rightarrow \infty D z f_{j, \lambda_{n}}$$

## 统计代写|离散时间鞅理论代写martingale代考|An Elliptic Equation for the Corrector Field

\left 的分隔符缺失或无法识别

，并考虑一个序列 \left 的分隔符缺失或无法识别

$$2 \sum_{z \in \Lambda}^{1}\left\langle p_{z} f_{j}(z), D_{z} g\right\rangle_{\mathbb{Q}}-\sum_{2 \in \Lambda^{s}}^{1}\left\langle q_{z} f_{j}(z), g\right\rangle_{\mathbb{Q}}=\sum_{z \in \Lambda}\left\langle p_{z} H_{j, z}, D_{z} g\right\rangle_{\mathbb{Q}}$$

## 统计代写|离散时间鞅理论代写martingale代考|The Energy Identity

$$\frac{1}{2} \sum_{z \in \Lambda}\left\langle p_{z} D_{z} F_{j}, D_{z} F_{j}\right\rangle_{\mathbb{Q}}$$

$$\frac{1}{2} \sum_{z \in \Lambda} \sum_{x \in \mathbb{Z}^{d}} p(x, x+z) \partial_{z} f_{j}(x) \partial_{z}\left[f_{j}(x) \varphi_{\varepsilon}(x)\right]$$

\$\$
$\backslash$ partial_{z} $\mathrm{fg}=\mathrm{f}(\mathrm{x}+\mathrm{z}) \backslash$ left( $\backslash$ partial_{z} $g \backslash$ right $)(\mathrm{x})+\mathrm{g}(\mathrm{x}) \backslash$ left( $\backslash$ partial_{z} f(right)(x),
thepreviousexpressionisequalto
$$\frac{1}{2} \sum_{z \in \Lambda} \sum_{x \in \mathbb{Z}^{d}} \dot{\varphi} \varepsilon(x+z) p(x, x+z) \partial z f_{j}(x)^{2} \quad+\frac{1}{2} \sum_{z \in \Lambda} \sum_{x \in \mathbb{Z}^{d}} \partial_{z} \varphi_{\varepsilon}(x) p(x, x+z) \partial_{z} f_{j}(x) f_{j}(x)$$

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