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\[0= \int_{0}^{T}\int_{\mathbb{R}^{d}}\{-u_{0}(\hat{Y}^{x,n}_{t}( \omega))\rho^{{}^{\prime}}(t)\eta(x)\] \[+b_{n}(t,x+B^{H}_{t}(\omega))\cdot(\nabla u_{0})(\hat{Y}^{x,n}_{ t}(\omega))^{T}\frac{\partial}{\partial x}\hat{Y}^{x,n}_{t}(\omega)\rho(t)\eta(x)\} dxdt\text{, for a.a. }\omega.\] \[\Big{(}\left(t,x\right)\mapsto\hat{Y}^{x,n_{j}}_{t}\Big{)}_{j\geq 1}\] \[\Big{(}\left(t,x\right)\mapsto\frac{\partial}{\partial x}\hat{Y}^{x,n_{j}}_{t} \Big{)}_{j\geq 1}\] \[Y(t,x,\omega)=\hat{Y}^{x}_{t}(\omega)\text{, }Y^{{}^{\prime}}(t,x,\omega)=\frac{ \partial}{\partial x}\hat{Y}^{x}_{t}(\omega)\] \[(\nabla u_{0})(\hat{Y}_{t}^{x}(\omega))^{T}\frac{\partial}{\partial x}\hat{Y }_{t}^{x}(\omega)=\nabla u(\omega,t,x)\quad(t,x)\text{-a.e.}\] \[\int_{0}^{T}\int_{\mathbb{R}^{d}}\{-u(\omega,t,x)\rho^{{}^{\prime}}(t)\eta(x) +b^{*}(t,x)\cdot\nabla u(\omega,t,x)\rho(t)\eta(x)\}dxdt=0\]
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2
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\[\|[e^{it\partial_{x}^{2}}\tilde{w}_{n}^{M}](0)\|_{L^{q}_{\mathbb{R}_{ t}}}\] \[\leq \|[e^{it\partial_{x}^{2}}w_{n}^{M}](0)\|_{L^{q}_{\mathbb{R}_{t}}}+ \sum_{j=1}^{M}\|[e^{it\partial_{x}^{2}}(-\mathrm{NLS}(-t_{n}^{j})\tilde{\phi}^{ j}+e^{-it_{n}^{j}\partial_{x}^{2}}\phi^{j})](0)\|_{L^{q}_{\mathbb{R}_{t}}}\] \[\leq \|[e^{it\partial_{x}^{2}}w_{n}^{M}](0)\|_{L^{q}_{\mathbb{R}_{t}}}+ \sum_{j=1}^{M}\|\mathrm{NLS}(-t_{n}^{j})\tilde{\phi}^{j}-e^{-it_{n}^{j} \partial_{x}^{2}}\phi^{j}\|_{\dot{H}^{\sigma_{c}}_{x}},\] \[\leq \|[e^{it\partial_{x}^{2}}w_{n}^{M}](0)\|_{L^{q}_{\mathbb{R}_{t}}}+ \sum_{j=1}^{M}\|\mathrm{NLS}(-t_{n}^{j})\tilde{\phi}^{j}-e^{-it_{n}^{j} \partial_{x}^{2}}\phi^{j}\|_{H^{1}_{x}}.\]
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3
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\[\frac{1}{2\tau}\left(\|\Phi^{k+1}-\Phi^{*}\|^{2}-\|\Phi^{k}-\Phi^{* }\|^{2}\right)\] \[\leq h\sum_{i=1}^{N-1}\Phi_{i}-h\sum_{i=0}^{N-1}(\Phi_{i})^{2}+ah \frac{\Phi_{m+1}(\Phi_{m+2}-\Phi_{m+1})}{\Phi_{m+2}-\Phi_{m+1}}+ah(1-2\Phi_{m+1})\] \[= h\sum_{i=1}^{N-1}\Phi_{i}+ah-ah\Phi_{m+1}-h\sum_{i=1}^{N-1}(\Phi _{i})^{2}\] \[= -\left[h\sum_{i=1}^{m}(\Phi_{i})^{2}+h(\Phi_{m+1}-a)^{2}+h\sum_{i =m+2}^{N-1}(\Phi_{i}-1)^{2}\right]+2h\sum_{i=1}^{m+1}\Phi_{i}-3ah\Phi_{m+1}+a^ {2}h\] \[\leq -\|\Phi^{k+1}-\Phi^{*}\|^{2}+2h\sum_{i=1}^{m+1}\Phi_{i}+h.\]
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4
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\[\mathbb{E}[H_{s}(\lambda_{t}-\mu)]-\mathbb{E}[H_{s}](\mathbb{E}[ \lambda_{t}]-\mu)\] \[=\mu\int_{0}^{s}\Phi(t-v)\left(1+\int_{v}^{s}\Psi(y-v)dy\right) \left(1+\int_{0}^{v}\Psi(w)dw\right)dv\] \[+\mu\int_{0}^{t}\int_{0}^{s\wedge u}\Phi(t-u)\Psi(u-v)\left(1+ \int_{0}^{v}\Psi(w)dw\right)\left(1+\int_{v}^{s}\Psi(y-v)dy\right)dvdu\] \[=\mu\int_{0}^{s}\Phi(t-v)\left(1+\int_{v}^{s}\Psi(y-v)dy\right) \left(1+\int_{0}^{v}\Psi(w)dw\right)dvd\] \[+\mu\int_{0}^{s}\int_{0}^{u}\Phi(t-u)\Psi(u-v)\left(1+\int_{0}^{v }\Psi(w)dw\right)\left(1+\int_{v}^{s}\Psi(y-v)dy\right)dvdu\] \[+\mu\int_{0}^{s}\Phi(t-v)\Psi(s-v)\left(1+\int_{0}^{v}\Psi(w)dw\right) \int_{v}^{s}\Phi(t-u)\Psi(u-v)dudv\] \[+\mu\int_{0}^{s}\Phi(t-v)\Psi(s-v)\left(1+\int_{0}^{v}\Psi(w)dw \right)dv\] \[+\mu\int_{0}^{t}\int_{0}^{s/u}\Phi(t-u)\Psi(u-v)\Psi(s-v)\left(1+ \int_{0}^{v}\Psi(v-w)dw\right)dv\] \[=\mathbb{E}\left[(\lambda_{s}-\mu)\right]\mathbb{E}\left[(\lambda _{t}-\mu)\right]\] \[+\mu\int_{0}^{s}\Phi(t-v)\Psi(s-v)\left(1+\int_{0}^{v}\Psi(w)dw \right)dv\] \[+\mu\int_{0}^{s}\Psi(s-v)\left(1+\int_{0}^{v}\Psi(v-w)dw\right) \int_{v}^{s}\Phi(t-u)\Psi(u-v)dudv\] \[+\mu\int_{0}^{s}\Psi(s-v)\left(1+\int_{0}^{v}\Psi(v-w)dw\right) \int_{s}^{t}\Phi(t-u)\Psi(u-v)dudv\] \[+\mu\int_{0}^{s}\Psi(s-v)\left(1+\int_{0}^{v}\Psi(v-w)dw\right) \int_{s}^{t}\Phi(t-u)\Psi(u-v)dudv\] \[=\mathbb{E}\left[(\lambda_{s}-\mu)\right]\mathbb{E}\left[(\lambda _{t}-\mu)\right]\] \[+\mu\int_{0}^{s}\Phi(t-v)\Psi(s-v)\left(1+\int_{0}^{v}\Psi(w)dw \right)dv\] \[+\mu\int_{0}^{s}\Psi(s-v)\left(1+\int_{0}^{v}\Psi(v-w)dw\right)\int_{v}^{ t}\Phi(t-u)\Psi(u-v)dudv\] \[=\mathbb{E}\left[(\lambda_{s}-\mu)\right]\mathbb{E}\left[(\lambda_ {t}-\mu)\right]\] \[+\mu\int_{0}^{s}\Phi(t-v)\Psi(s-v)\left(1+\int_{0}^{v}\Psi(w)dw \right)dv\] \[+\mu\int_{0}^{s}\Psi(s-v)\left(1+\int_{0}^{v}\Psi(v-w)dw\right) \int_{0}^{t-v}\Phi(t-v-x)\Psi(x)dxdv\] \[=\mathbb{E}\left[(\lambda_{s}-\mu)\right]\mathbb{E}\left[(\lambda _{t}-\mu)\right]\] \[+\mu\int_{0}^{s}\Psi(s-v)\Psi(t-v)\left(1+\int_{0}^{v}\Psi(v-w)dw \right)dv.\]
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5
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\[z_{r}(\bm{Q}^{\text{RS}}) =\frac{1}{2}\sum_{\sigma^{0},\cdots,\sigma^{r}\in\{\pm 1\}}\exp \Big{[}\beta\lambda b\sum_{a=1}^{r}\sigma^{0}\sigma^{a}+\beta^{2}q\sum_{1\leq a <b\leq r}\sigma^{a}\sigma^{b}\Big{]}\] \[=\sum_{\sigma^{1}\cdots,\sigma^{r}\in\{\pm 1\}}\exp\Big{[}\beta \lambda b\sum_{a=1}^{r}\sigma^{a}+\frac{\beta^{2}q}{2}\Big{(}\sum_{a=1}^{r} \sigma^{a}\Big{)}^{2}-\frac{\beta^{2}qr}{2}\Big{]}\] \[=\mathbb{E}\Big{[}\sum_{\sigma^{1},\cdots,\sigma^{r}}\exp\Big{[} \beta\lambda b\sum_{a=1}^{r}\sigma^{a}+\beta\sqrt{q}g\sum_{a=1}^{r}\sigma^{a} -\frac{\beta^{2}qr}{2}\Big{]}\Big{]}\] \[=\exp\Big{[}-\frac{\beta^{2}qr}{2}\Big{]}\mathbb{E}\Big{[}(2 \text{cosh}\beta(\lambda b+\sqrt{q}g))^{r}\Big{]},\]
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6
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\[g\left(n_{\Sigma_{\tau}},n_{\Sigma_{\tau}}\right) =\begin{cases}-1-\dfrac{2Mr(r^{2}+a^{2})}{\Delta\rho^{2}}+\dfrac {\Delta}{\rho^{2}}\left(\dfrac{df}{dr}\right)^{2}\,,&r\geq 9M/4\\ -1-\dfrac{2Mr}{\rho^{2}}-\dfrac{4Mr}{\rho^{2}}\dfrac{df}{dr}+\dfrac{\Delta}{ \rho^{2}}\left(\dfrac{df}{dr}\right)^{2}\,,&r\leq 15M/8\end{cases}\,,\] \[g\left(n_{\Sigma_{\tau}},L\right) =\begin{cases}1-\dfrac{\Delta}{r^{2}+a^{2}}\dfrac{df}{dr}\,,&r \geq 9M/4\,,\\ 1+\dfrac{2Mr}{r^{2}+a^{2}}-\dfrac{\Delta}{r^{2}+a^{2}}\dfrac{df}{dr}\,,&r \leq 15M/4\,,\end{cases}\,,\] \[g\left(n_{\Sigma_{\tau}},\dfrac{r^{2}+a^{2}}{\Delta}\underline{L}\right) =\begin{cases}\dfrac{r^{2}+a^{2}}{\Delta}+\dfrac{df}{dr}\,,&r \geq 9M/4\,,\\ 1+\dfrac{df}{dr}\,,&r\leq 15M/8\,,\end{cases}\,,\] \[\det g_{\Sigma_{\tau}} =\rho^{2}\sin^{2}\theta\begin{cases}w^{-1}-\Delta\left(\dfrac{ df}{dr}\right)^{2}-a^{2}\sin^{2}\theta\,,&r\geq 9M/4\,,\\ \rho^{2}+2Mr\left(1+2\dfrac{df}{dr}\right)-\Delta\left(\dfrac{df}{dr}\right)^ {2}\,,&r\leq 15M/8\,,\end{cases}\,.\]
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7
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\[\left(\begin{array}{c}Z\\ \Phi\end{array}\right)_{t} =-\mathbf{A}_{kl}\left(\begin{array}{c}Z\\ \Phi\end{array}\right)+2\sum_{i,j=1}^{N_{kl}}\lambda^{kl}_{j}\left\langle \left(\begin{array}{c}\mathcal{L}_{kl}\mathbb{D}^{1}_{\gamma^{kl}_{i}}(\psi^{kl }_{i})\\ \mathbb{D}^{2}_{\gamma^{kl}_{i}}(\psi^{kl}_{i})\end{array}\right),\left( \begin{array}{c}Z^{kls}_{j}\\ \Phi^{kls}_{j}\end{array}\right)\right\rangle\left(\begin{array}{c}Z^{kl}_{j} \\ \Phi^{kl}_{j}\end{array}\right)\] \[+\sum_{i=1}^{N_{kl}}\gamma^{kl}_{i}\left(\begin{array}{c} \mathcal{L}_{kl}\mathbb{D}^{1}_{\gamma^{kl}_{i}}(\psi^{kl}_{i})\\ \mathbb{D}^{2}_{\gamma^{kl}_{i}}(\psi^{kl}_{i})\end{array}\right)-\sum_{i=1}^{N_{ kl}}\left(\begin{array}{c}\mathcal{L}_{kl}\mathbb{D}^{1}_{\gamma^{kl}_{i}}(\psi^{kl }_{i})\\ \mathbb{D}^{2}_{\gamma^{kl}_{i}}(\psi^{kl}_{i})\end{array}\right)_{t}.\]
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8
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\[\frac{\mathrm{d}}{\mathrm{d}t}\|g\|_{\dot{H}^{\frac{1}{2}}}^{2} +C(M)\bigg{[}\log\left(4+||f_{1}||_{2,\frac{1}{3}}\right)^{-\frac{1 }{3}}+\log\left(4+||f_{2}||_{2,\frac{1}{3}}\right)^{-\frac{1}{3}}\bigg{]}\,\|g\|_ {\dot{H}^{1}}^{2}\] \[\lesssim_{M}\log\left(4+\|f_{1}\|_{2,\frac{1}{3}}\right)^{-\frac{ 1}{3}}\|f_{1}\|_{2,\frac{1}{3}}\,\|g\|_{\dot{H}^{\frac{1}{2}}}\,\|g\|_{\dot{H} ^{1}}\] \[+\log\left(4+\|f_{2}\|_{2,\frac{1}{3}}\right)^{-\frac{1}{3}}\|f_{ 2}\|_{2,\frac{1}{3}}^{\frac{1}{2}}\,\|g\|_{\dot{H}^{\frac{1}{2}}}^{\frac{1}{2}} \,\|g\|_{\dot{H}^{1}}^{\frac{3}{2}}\] \[+\log\left(4+\|f_{2}\|_{2,\frac{1}{3}}\right)^{-\frac{1}{3}}\|f_{ 2}\|_{2,\frac{1}{3}}\,\|g\|_{\dot{H}^{\frac{1}{2}}}\,\|g\|_{\dot{H}^{1}}\] \[+\log\left(4+\|f_{1}\|_{2,\frac{1}{3}}\right)^{-\frac{1}{3}}\|f_{ 1}\|_{2,\frac{1}{3}}^{\frac{1}{2}}\,\|g\|_{\dot{H}^{\frac{1}{2}}}^{\frac{1}{2} }\,\|g\|_{\dot{H}^{1}}^{\frac{3}{2}}\,.\]
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9
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\[(A_{H,2})_{i,j} = \int_{\Omega}\zeta(u_{H}+\alpha\widetilde{u}_{h})^{2}\phi_{i,H} \phi_{j,H}d\Omega\] \[= \int_{\Omega}\zeta\big{(}(u_{H})^{2}+2\alpha u_{H}\widetilde{u}_ {h}+\alpha^{2}(\widetilde{u}_{h})^{2}\big{)}\phi_{i,H}\phi_{j,H}d\Omega\] \[= \int_{\Omega}\zeta(u_{H})^{2}\phi_{i,H}\phi_{j,H}d\Omega+2\alpha \int_{\Omega}\zeta\widetilde{u}_{h}u_{H}\phi_{i,H}\phi_{j,H}d\Omega\] \[\quad+\alpha^{2}\int_{\Omega}\zeta(\widetilde{u}_{h})^{2}\phi_{i,H}\phi_{j,H}d\Omega\] \[:= (A_{H,2,1})_{i,j}+2\alpha(A_{H,2,2})_{i,j}+\alpha^{2}(A_{H,2,3})_ {i,j}.\]
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10
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\[\begin{split}&\hat{\mathbb{E}}[(\int_{0}^{T}n|(Y_{s}^{n})^{-}|^{ \alpha-1}(Y_{s}^{m})^{-}ds)^{\alpha^{\prime}}]\\ &\leq\hat{\mathbb{E}}[\sup_{s\in[0,T]}\{|(Y_{s}^{n})^{-}|^{( \alpha-2)\alpha^{\prime}}|(Y_{s}^{m})^{-}|^{\alpha^{\prime}}\}(\int_{0}^{T}n(Y _{s}^{n})^{-}ds)^{\alpha^{\prime}}]\\ &\leq(\hat{\mathbb{E}}[\sup_{s\in[0,T]}|(Y_{s}^{n})^{-}|^{( \alpha-2)\alpha^{\prime}p}])^{\frac{1}{p}}(\hat{\mathbb{E}}[\sup_{s\in[0,T]}|(Y _{s}^{m})^{-}|^{\alpha^{\prime}q}])^{\frac{1}{q}}(\hat{\mathbb{E}}[(\int_{0}^{ T}n(Y_{s}^{n})^{-}ds)^{\alpha^{\prime}r}])^{\frac{1}{r}}\\ &\leq C(\hat{\mathbb{E}}[\sup_{s\in[0,T]}|(Y_{s}^{m})^{-}|^{ \alpha^{\prime}q}])^{\frac{1}{q}},\end{split}\]
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