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SCIENTIA SINICA Mathematica, Volume 49 , Issue 3 : 339(2019) https://doi.org/10.1360/N012018-00053

A brief introduction to KPZ equation and KPZ universality

More info
  • ReceivedMar 9, 2018
  • AcceptedOct 16, 2018
  • PublishedMar 13, 2019

Abstract


Funded by

中国科学院数学与系统科学研究院华罗庚数学科学中心

中国科学院随机复杂结构与数据科学重点实验室(2008DP173182)

国家自然科学基金(11431014,11688101 和 11801546)


Supplement

Appendix

KPZ 方程的尺度变换

我们仅考虑$d=1$的情形. 由 13, KPZ 方程一般可写作 \begin{equation} \frac{\partial h}{\partial t}=\nu \Delta h + \frac{\lambda}{2}(\nabla h)^2+\eta(t,x), \tag{61}\end{equation} 其中, $\nu,~\lambda~\neq~0,~$ $D>0$, $\eta$是时空白噪声, 即${\rm~E}\eta(t,x)=0,$ $~{\rm~E}[\eta(t,x)\eta(t',x')]=D\delta(t-t')\delta(x-x')$. 也就是说, 参数组$(\nu,~\lambda,~D)$决定了 KPZ 方程的具体形式. 本节 将说明如下事实: 给定另一组参数$(\nu',~\lambda',~D')$ ($\nu',\lambda'\neq~0,~$ $D'>0$), 由这组参数确定的 KPZ 方程可以由 61通过合适的尺度变换得到.

事实上, 令 \begin{equation} h_\zeta(t,x):=\zeta^a h(\zeta^{-b}t, \zeta^{-1}x), \tag{62}\end{equation} 其中, $\zeta,~a,~b>0$是待定的参数. 不难验证, $h_\zeta$满足如下方程: \begin{equation}\frac{\partial h_\zeta}{\partial t}=(\zeta^{2-b}\nu) \Delta h_\zeta + \bigg (\zeta^{2-a-b}\frac{\lambda}{2}\bigg )(\nabla h_\zeta)^2+\eta_\zeta(t,x), \end{equation} 其中, $\eta_\zeta(t,x)=\zeta^{a-b}\eta(\zeta^{-b}t,\zeta^{-1}x)$. 特别地, $\eta_\zeta$仍然是时空白噪声, 并且${\rm~E}\eta_\zeta=0$, 以及 \begin{equation}\begin{aligned} {\rm E}\eta_\zeta(t,x)\eta_\zeta(t',x')&=\zeta^{2a-2b}{\rm E}\eta(\zeta^{-b}t,\zeta^{-1}x)\eta(\zeta^{-b}t',\zeta^{-1}x') \\ &=\zeta^{2a-2b}D\delta(\zeta^{-b}(t-t'))\delta(\zeta^{-1}(x-x')) \\ &=\zeta^{2a-b+1}D\delta(t-t')\delta(x-x'). \end{aligned} \end{equation} 上式中的第三个等式用到了 Dirac 函数的如下性质: $\delta(ct-ct')=c^{-1}\delta(t-t')$ ($c\neq~0$). 于是, 只需要取合适的$\zeta$、$a$和$~b$使得 \begin{equation} \zeta^{2-b}=\frac{\nu'}{\nu}, \zeta^{2-a-b}=\frac{\lambda'}{\lambda}, \zeta^{2a-b+1}=\frac{D'}{D}, \tag{63}\end{equation} 如上定义的$h_\zeta$即满足由参数$(\nu',\lambda',D')$确定的 KPZ 方程.

我们再来观察由上述事实得到的一个简单推论. 在 62中考虑$3:2:1$的尺度变换, 即取$b=3/2,$ $~a=1/2$. 此时, 63给出$\nu'=\zeta^{1/2}\nu,~$ $\lambda'=\lambda,~$ $D'=\zeta^{1/2}D$. 也就是说, KPZ 方程并不满足$3:2:1$的尺度不变性.

Cole-Hopf 变换

本节解释针对一维 KPZ 方程的 Cole-Hopf 变换, 它将 KPZ 方程转化为随机热方程, 从而实现了方程的 “线性化". 为了简便起见, 并与 第sect. 3 节一致, 我们仍然假设$\nu=1/2,~$ $\lambda=D=1$.

如 第sect. 3.1 小节所解释, 时空白噪声可理解为柱 Wiener 过程关于时间的 Itô 型形式导数. 于是, 若忽略空间变量造成的奇异性 (空间变量的奇异性可利用磨光技巧来处理), 21描述的函数$h(t,x)$关于时间变量$t$差不多是一个半鞅. 进而, 在 Cole-Hopf 变换下, 我们应该利用 Itô 公式给出 \begin{equation}Y(t,x)=\exp\{h(t,x)\} \end{equation} 满足的方程. 具体地, 我们将 21写作 \begin{equation} \partial_t h= \bigg (\frac{1}{2}\Delta h +\frac{1}{2}(\nabla h)^2-\infty \bigg )dt+dW_t, \tag{64}\end{equation} 其中, $W_t$是$L^2(\mathbb{R})$上的柱 Wiener 过程, 再利用 Itô 公式, 可以得到 ($Y_t:=Y(t,\cdot)$) \begin{equation}d Y_t=Y_t \partial_t h + \frac{1}{2}Y_t \mathrm{tr}(I)dt, \end{equation} 其中, $I$是柱 Wiener 过程的协方差算子, 即$L^2(\mathbb{R})$上的恒等算子, $\mathrm{tr}(I)$表示$I$的迹, 它是发散的. 将 64代入上式, 即有 \begin{equation}d Y_t=\frac{1}{2}\Delta Y_tdt +Y_t \bigg (\frac{1}{2}\mathrm{tr}(I)-\infty\bigg )dt +Y_tdW_t. \end{equation} 由此不难看出, Cole-Hopf 变换为了将 KPZ 方程转化为线性方程, 并且 (在形式上)保证线性化后的方程有意义, 还需对 21中的重整项作出如下诠释: \begin{equation}\infty=\frac{1}{2}\mathrm{tr}(I). \end{equation} 特别地, $Y_t$满足可乘噪声驱动的随机热方程 22.

Wick 积

Wick 积的一般定义

Wick 积 是物理学家 Wick 在 文献[54]中提出的一种 (任意阶矩均有限的)随机变量之间的运算. 给定某个概率空间$(\Omega,~\mathcal{F},~\mathbf{P})$, 对于$n\geq~0$和$\Omega$上$n$个任意阶可积的随机变量$X_1,~X_2,\ldots,~X_n$ (${\rm~E}X_i^p<\infty$, $1\leq~i\leq~n,~$ $~\forall\,~p\geq~1$), 它们的 Wick 积记作$\langle~X_1,~X_2,\ldots,~X_n\rangle$ (有些文献中也将 Wick 积记作$:X_1X_2\cdots~X_n\!:$, 如 文献[55]), 它按照如下的递归方式给出.

Definition 4. 令$\mathcal{V}$是$(\Omega,~\mathcal{F},~\mathbf{P})$上任意阶矩均有限的随机变量全体. 对于$n\geq~0$, \begin{equation}\langle\cdot \rangle_n: \mathcal{V}^n\rightarrow \mathcal{V}, (X_1,\ldots, X_n)\mapsto \langle X_1,\ldots, X_n\rangle_n \end{equation} 是按照如下的递归方式定义的泛函:

(1) $\langle~\cdot~\rangle_0=1$;

(2) 假设$\langle~\cdot~\rangle_{n-1}$已经定义, 对于$X_1,\ldots,~X_n\in~\mathcal{V}$, $\langle~X_1,~X_2,\ldots,~X_n\rangle_n$由${\rm~E}\langle~X_1,\ldots,~X_n~\rangle_n=0$和 \begin{equation}\frac{\partial \langle X_1,\ldots , X_n\rangle_n}{\partial X_i}=\langle X_1,\ldots X_{i-1}, X_{i+1},\ldots X_n\rangle_{n-1}, 1\leq i\leq n \end{equation} 确定.

称这族泛函$\{\langle\cdot~\rangle_n:~n\geq~0\}$为 Wick 积. 一般地, 在不引起混淆时, 我们将省略 Wick 积中用来标记随机变量数量的下角标$n$.

Remark 15. 为了方便起见, 记 $$X'^{n}:=\underbrace{X, \ldots, X}_{n \text{个}},$$ 于是, $$\langle X'^n\rangle=\langle \underbrace{X, \ldots, X}_{n \text{个}}\rangle.$$ 特别地, 利用定义 4中的第二条, 我们可以得到 \begin{equation} \frac{d\langle X'^n\rangle}{dX}=n\langle X'^{n-1}\rangle. \tag{65}\end{equation}

不难证明(参见文献[56]), 按上述递归方式给出的 Wick 积是良定的. 作为例子, 容易计算前两项 Wick 积是 \begin{equation} \langle X\rangle=X-{\rm E}X, \langle X, Y\rangle=XY-X{\rm E}Y-Y{\rm E}X+2{\rm E}X {\rm E}Y-{\rm E}(XY). \tag{66}\end{equation} 并且, \begin{equation}\mathcal{V}^n\rightarrow \mathcal{V}, (X_1,\ldots, X_n)\mapsto \langle X_1,\ldots, X_n\rangle \end{equation} 是$n$元对称多线性泛函.

Lemma 1. ([56] 关于 Wick 积的以下性质成立:

(1) 对于任意$\alpha,~\beta~\in~\mathbb{R}$, 有$\langle~\alpha~X_1+\beta~\tilde{X}_1,~X_2,\ldots,~X_n\rangle=\alpha~\langle~X_1,~X_2,\ldots,~X_n\rangle+\beta~\langle~\tilde{X}_1,~X_2,\ldots,~X_n\rangle$;

(2) 对于$i<j$, 有$\langle~X_1,\ldots~X_{i},\ldots,~X_{j},\ldots~X_n\rangle=\langle~X_1,~\ldots~X_{j},\ldots,~X_{i},\ldots~X_n\rangle$.

需要注意的是, 对于 Wick 积, 结合律通常是不成立的. 也就是说, 一般地, 若$X,~Y,~Z\in~\mathcal{V}$, 则 \begin{equation}\langle X, Y, Z\rangle \neq \langle \langle X, Y\rangle, Z\rangle. \end{equation} 但在附录 sect. 3.3中我们将发现, 如果限定在零均值的 Gauss 型随机变量上, 那么利用 Wick 积可以诱导出一种真正的乘法运算, 它的基础在于对独立随机变量的如下观察. 由 66不难知道, 若$X$和$Y$独立, 那么$\langle~X,~Y\rangle~=\langle~X\rangle~\langle~Y\rangle$. 更一般地, 如下命题成立.

Lemma 2. ([56] 如果$X_1,\ldots,~X_k\in~\mathcal{V}$两两独立, 那么,对任意$n_1,\ldots,~n_k\in~\mathbb{N}$,有 \begin{equation}\langle X'^{n_1}_1, X'^{n_2}_2,\ldots, X'^{n_k}_k\rangle = \langle X'^{n_1}_1\rangle \langle X'^{n_2}_2\rangle \cdots \langle X'^{n_k}_k\rangle. \end{equation}

Gauss 型随机变量的 Wick 积

常用的是 Gauss 型随机变量之间的 Wick 积. 同时, 为了简便起见, 以下仅考虑均值为$0$的随机变量. 记$\mathcal{V}_G^0$是均值为$0$的 Gauss 随机变量全体. 显然, $\mathcal{V}_G^0\subset~\mathcal{V}$.

对于满足标准正态分布的随机变量$X$和$Y$, 由 6566, 我们不难证明 \begin{equation} \begin{aligned} &\langle X\rangle=X, \langle X, Y\rangle=XY-{\rm E}(XY), \\ & \langle X'^{n}\rangle =H_n(X), \end{aligned} \tag{67}\end{equation} 其中, $H_n(\xi)=(-1)^n\mathrm{e}^{\xi^2/2}\frac{d^n}{d\xi^n}\mathrm{e}^{-\xi^2/2}$是 Hermite 多项式. 特别地, 沿用前述符号, 记$\{\eta(t,x):~t\geq~0,~x$ $\in~\mathbb{R}^d\}$是时空白噪声 ($D=1$). 对于满足$\|f\|_{L^2([0,\infty)\times~\mathbb{R}^d)}=1$的$f\in~\mathcal{S}$, \begin{equation}X_f:=\eta(f)=\int f(t,x)\eta(t,x)dtdx \end{equation} 符合标准正态分布, 即$X_f\sim~N(0,1)$. 利用 67和 Wiener-Itô 公式(参见文献[57]), 我们可以得到$X_f$的$n$次 Wick 幂的如下表示: \begin{equation}\begin{aligned} \langle X'^n_f \rangle&=H_n(X_f) \\ &=n!\int_{\Lambda_n}\int_{(\mathbb{R}^d)^n}f(t_1,y_1)\cdots f(t_n, y_n)\eta(t_1,y_1)\cdots \eta(t_n,y_n)dy_1\cdots dy_ndt_1\cdots dt_n, \end{aligned}\end{equation} 其中, $\Lambda_n:=\{(t_1,\ldots,~t_n):~0<~t_1<t_2<\cdots<t_n\}$. 注意, 根据 第sect. 3.1小节, 上式中的$\eta(t_i,y_i)dt_i$可以理解为$dW_{t_i}(y_i)$ ($W$是柱 Wiener 过程). 由于 Wick 积满足多线性, 不难验证, 对于任意$f\in~\mathcal{S}$ (未必满足$\|f\|_{L^2([0,\infty)\times~\mathbb{R}^d)}=1$)仍然具有上述表示 (注意, $\langle~X'^n_f~\rangle=H_n(X_f)$不再成立了). 也就是说, 我们可以得到如下命题.

Proposition 1. [57] 对于任意$f\in~\mathcal{S}$, $X_f:=\eta(f)$的$n$次 Wick 积满足如下表示: \begin{equation}\langle X'^{n}_f\rangle=n!\int_{\Lambda_n}\int_{(\mathbb{R}^d)^n}f(t_1,y_1)\cdots f(t_n, y_n)\eta(t_1,y_1)\cdots \eta(t_n,y_n)dy_1\cdots dy_ndt_1\cdots dt_n, \end{equation} 其中, $\Lambda_n:=\{(t_1,\ldots,~t_n):~0<~t_1<t_2<\cdots<t_n\}$.

Remark 16. 需要指出的是, \begin{equation}\eta^{\otimes n}(\boldsymbol{t},\boldsymbol{y}):=\eta(t_1,y_1)\cdots \eta(t_n,y_n) \end{equation} 是作用在 $$\underbrace{\mathcal{S}\times \cdots \times \mathcal{S}}_{n}$$ 上的广义函数, 但利用 Itô 等距的思想, 很容易将$\eta^{\otimes~n}$的作用函数空间推广到$L^2(\Lambda_n\times~(\mathbb{R}^d)^n)$.

下面来考虑$\eta(t,x)$与$\eta(t',x')$的 Wick 积. 由于时空白噪声$\eta$在 第sect. 3.1小节被解释为广义函数, 因而有关$\eta$的运算需要作用到合适的测试函数之后才是可理解的 (例如, ${\rm~E}\eta(t,x)=0$需要理解为对任意$f\in~\mathcal{S}$, 有${\rm~E}~(\eta(f)~)=0$). 于是, Wick 积$\langle~\eta(t,x),~\eta(t',x')\rangle$需要将两个$\eta$分别作用到某个测试函数后才能定义. 也就是说, $\langle~\eta^{\otimes~2}\rangle:=\langle~\eta(t,x),~\eta(t',x')\rangle\in~(\mathcal{S}\otimes~\mathcal{S})'$, 并且对$f\otimes~g\in~\mathcal{S}\otimes~\mathcal{S}$,有 \begin{equation} \langle \eta^{\otimes 2}\rangle(f\otimes g):=\langle X_f, X_g\rangle, \tag{68}\end{equation} 其中, $X_f=\eta(f),~$ $X_g=\eta(g)$. 当然可以直接计算$\langle~X_f,~X_g\rangle$, 但将$f\otimes~g$对称化之后, 我们可以给出 68更简洁的表达式. 具体来说, 令$S(f\otimes~g):=(f\otimes~g+g\otimes~f)/2$, 那么由命题 1可以得到 \begin{equation}\langle \eta^{\otimes 2}\rangle (S(f\otimes g) )=2\int_{\Lambda_2}\int_{(\mathbb{R}^d)^2} S(f\otimes g)(\boldsymbol{t},\boldsymbol{y})\eta^{\otimes 2}(\boldsymbol{t},\boldsymbol{y})d\boldsymbol{y}d\boldsymbol{t}. \end{equation} 类似地, 利用 Itô 等距的思想, 可以将 68中的测试函数类$\mathcal{S}\otimes~\mathcal{S}$推广到$L^2(\Lambda_2\times~(\mathbb{R}^d)^2)$. 一般地, 利用 Wiener-Itô 公式和 Itô 等距, 我们可以得到时空白噪声的$n$次 Wick 积的如下描述.

Proposition 2. 对$n\geq~1$, 记$\langle~\eta^{\otimes~n}\rangle$是$\eta$的$n$次 Wick 幂,则对于$L^2(\Lambda_n\times~(\mathbb{R}^d)^n)$中的对称函数$\varphi$,有 \begin{equation}\langle \eta^{\otimes n}\rangle(\varphi)=n!\int_{\Lambda_n}\int_{(\mathbb{R}^d)^n} \varphi(\boldsymbol{t},\boldsymbol{y})\eta^{\otimes n}(\boldsymbol{t},\boldsymbol{y})d\boldsymbol{y}d\boldsymbol{t}. \end{equation}

最后, 我们来计算 CDRP 的配分函数 58. 注意 Wick 指数函数按照通常意义定义, 即 \begin{equation}\langle\exp\rangle\{X\}:=\sum_{n\geq 0}\frac{\langle X'^{n}\rangle}{n!}. \end{equation} 记$\mathbf{p}_n(\mathbf{s},\boldsymbol{y}):=p_{s_1,\ldots,~s_n}(y_1,\ldots,~y_n)$表示Brown桥的$n$步转移概率密度函数, 即 \begin{equation}\mathbf{P}_\mathrm{bb}(\omega_\mathrm{bb}(s_1)\in dy_1,\ldots, \omega_\mathrm{bb}(s_n)\in dy_n)=p_{s_1,\ldots, s_n}(y_1,\ldots, y_n)dy_1\cdots dy_n. \end{equation} 不难验证, $\mathbf{p}_n\in~L^2(\Lambda_n(t)\times~\mathbb{R}^n)$是对称函数, 并且$\sum_{n=0}^\infty~\|\mathbf{p}_n\|_{L^2(\Lambda_n\times~\mathbb{R}^n)}^2<\infty$. 于是, \begin{equation}\begin{aligned} Z(t,x)&={\rm E}_\mathrm{bb} \bigg (\langle\exp\rangle \bigg \{-\int_0^t \eta(s, b_s(\omega_\mathrm{bb}))ds\bigg \}\bigg ) \\ &=\sum_{n=0}^\infty \frac{(-1)^n}{n!}{\rm E}_\mathrm{bb} \bigg (\bigg \langle \bigg \{\int_0^t \eta(s, b_s(\omega_\mathrm{bb}))ds\bigg \}'^{n}\bigg \rangle \bigg ) \\ &=\sum_{n=0}^\infty \frac{(-1)^n}{n!}\int_{\mathbb{R}^n} \bigg \langle\int_0^t \eta(s_1,y_1)ds_1, \ldots, \int_0^t \eta(s_n, y_n)ds_n\bigg \rangle \mathbf{p}_n(\mathbf{s}, \boldsymbol{y})d\boldsymbol{y} \\ &=\sum_{n=0}^\infty \frac{(-1)^n}{n!} \langle\eta^{\otimes n}\rangle (\mathbf{p}_n) \\ &=\sum_{n=0}^\infty \int_{\Lambda_n(t)} \int_{\mathbb{R}^n} (-1)^n\mathbf{p}_n(\mathbf{s},\boldsymbol{y})\eta^{\otimes n}(\mathbf{s}, \boldsymbol{y}) d\boldsymbol{y} d\mathbf{s}. \end{aligned} \end{equation} 这实际上就是$Z(t,x)$在$L^2(\Omega,~\mathbf{P})$中的 Wiener-Chaos 展开 ($\Omega$是定义时空白噪声$\eta$的概率空间).

白噪声分析

在白噪声分析中, 我们将概率空间$(\Omega,~\mathcal{F},~\mathbf{P})$取作某个抽象 Wiener 空间$(B,~H,~\mu)$. 特别地, 文献[58] 在白噪声框架下选取$B=\mathcal{S}'(\mathbb{R}^d),~$ $H=L^2(\mathbb{R}^d),~$ $B'=\mathcal{S}(\mathbb{R}^d)$, 而 文献[28]在时空白噪声的框架下选取$B=\mathcal{S}'([0,\infty)\times~\mathbb{R}^d),$ $~H=L^2([0,\infty)\times\mathbb{R}^d),~$ $~B'=\mathcal{S}([0,\infty)\times\mathbb{R}^d)$. 这里为了简便起见, 我们不详细说明抽象 Wiener 空间的具体形式, 但取定 \begin{equation}\{\varphi_n :n\geq 1\}\subset B' \overset{\iota}{\hookrightarrow} H \end{equation} 使得 $\{\iota~(\varphi_n):~n\geq~1~\}$是$H$中的一族标准正交基. 特别地, $\{\varphi_n:~n\geq~1\}$是$(B,~\mu)$上相互独立的标准 Gauss 型随机变量族. 定义指标集 \begin{equation}J:=\{\alpha=(\alpha_n)_{n\geq 1}: \alpha_n\in \mathbb{N}, \text{ 最多有限个} \alpha_n\neq 0\}. \end{equation} 根据引理 2, 对于$\alpha\in~J$,有 \begin{equation} \langle \varphi'^\alpha\rangle:= \langle \varphi'^{\alpha_1}_1, \varphi'^{\alpha_2}_2,\ldots \rangle = \langle \varphi'^{\alpha_1}_1\rangle \langle \varphi'^{\alpha_2}_2\rangle \cdots =\prod_{n\geq 1}H_{\alpha_n}(\varphi_n). \tag{69}\end{equation} 令$\mathbf{H}_\alpha(\omega):=\langle~\varphi'^\alpha\rangle(\omega)=\prod_{n\geq~1}H_{\alpha_n}(\varphi_n(\omega))$, $\omega\in~B$. 进而, $\{\sqrt{\alpha!}^{-1}\mathbf{H}_\alpha:~\alpha\in~J\}$构成了$L^2(B,\mu)$的一组标准正交基.

我们的讨论将在 Kondratiev 分布空间$(\mathcal{S}^{-1})$上进行展开, 它包含$L^2(B,\mu)$, 其中的分布函数可以写作$\{\sqrt{\alpha!}^{-1}\mathbf{H}_\alpha:~\alpha\in~J\}$的线性组合: \begin{equation}L^2(B,\mu)\subset (\mathcal{S}^{-1}) \subset \bigg \{\sum_{\alpha\in J}a_\alpha \mathbf{H}_\alpha: a_\alpha\in \mathbb{R}\bigg \}. \end{equation} $(\mathcal{S}^{-1})$的具体形式参见文献[28,58], 引入它的意义在于 Kondratiev 分布空间关于下面要定义的 (由 Wick 积诱导的)乘法是封闭的, 而且时空白噪声$\eta(t,x)\in~(\mathcal{S}^{-1})$.

Remark 17. 在考察时空白噪声$\eta(t,x)$时, 概率空间是定义时空白噪声的概率空间, 故而$(B,H,\mu)$的选取与 文献[28]一致. 此时, 记$f_n:=\iota(\varphi_n)$, $n\geq~1$. 它们构成了$L^2([0,\infty)\times~\mathbb{R}^d)$的一组标准正交基. 反过来, \begin{equation}\varphi_n=\eta(f_n)=X_{f_n}. \end{equation} 于是, $\eta$可以写作(参见文献 [28]) \begin{equation} \eta(t,x)=\sum_{n\geq 1}f_n(t,x)\varphi_n=\sum_{n\geq 1}f_n(t,x)\mathbf{H}_{e_n} \in (\mathcal{S}^{-1}) , \tag{70}\end{equation} 其中, $e_n=(0,\ldots,~1,~0,\ldots)$. 显然, 由于 \begin{equation}\bigg \|\sum_{n\geq 1}f_n(t,x)\mathbf{H}_{e_n}\bigg \|^2_{L^2(B,\mu)} = \sum_{n\geq 1} f_n(t,x)^2, \end{equation} 而后者并不收敛, 故$\eta(t,x)\notin~L^2(B,\mu)$.

在$(\mathcal{S}^{-1})$上, 我们可以按照如下方式定义一种乘法 “$\diamond$”, 它实际上是由 Wick 积$\langle\cdot~\rangle$诱导的, 在有些文献中 (如 文献[28,58]等)也直接称 “$\diamond$” 为 Wick 积或 Wick 乘法.

Definition 5. 映射 \begin{equation}\begin{aligned} \diamond: &(\mathcal{S}^{-1})\times (\mathcal{S}^{-1})\rightarrow (\mathcal{S}^{-1}), \\ & \bigg (\sum_{\alpha\in J}a_\alpha \mathbf{H}_\alpha, \sum_{\beta\in J}b_\beta \mathbf{H}_\beta\bigg )\mapsto \bigg (\sum_{\alpha\in J}a_\alpha \mathbf{H}_\alpha\bigg )\diamond \bigg (\sum_{\beta\in J}b_\beta \mathbf{H}_\beta\bigg )=:\sum_{\alpha, \beta\in J}a_\alpha b_\beta \mathbf{H}_{\alpha+\beta} \end{aligned}\end{equation} 定义了$(\mathcal{S}^{-1})$上的一种乘法. 在不引起混淆时, 我们也称运算$\diamond$为 Wick 积或 Wick 乘法.

根据 文献[58], $(\mathcal{S}^{-1})$关于上述运算$\diamond$封闭, 换句话讲, 上述定义是良定的. 另外, 下面的性质显然成立.

Lemma 3. ([58] 对于$\mathbf{F}_1,\mathbf{F}_2,\mathbf{F}_3\in~(\mathcal{S}^{-1})$, $c\in~\mathbb{R}$, 下述性质成立: \begin{equation}\begin{aligned} &\mathbf{F}_1\diamond \mathbf{F}_2=\mathbf{F}_2\diamond \mathbf{F}_1, (c\mathbf{F}_1)\diamond \mathbf{F}_2=c(\mathbf{F}_1\diamond \mathbf{F}_2), \\ & (\mathbf{F}_1+\mathbf{F}_2)\diamond \mathbf{F}_3=\mathbf{F}_1\diamond \mathbf{F}_3+\mathbf{F}_2\diamond \mathbf{F}_3, \\ & (\mathbf{F}_1\diamond \mathbf{F}_2)\diamond \mathbf{F}_3=\mathbf{F}_1\diamond (\mathbf{F}_2\diamond \mathbf{F}_3). \end{aligned} \end{equation}

不难看出, 定义 5最本质的设定在于 \begin{equation}\mathbf{H}_\alpha \diamond \mathbf{H}_\beta:=\mathbf{H}_{\alpha+\beta}. \end{equation} 根据 69, 它可以写作 \begin{equation}\langle \varphi'^\alpha\rangle \diamond \langle \varphi'^\beta\rangle=\langle \varphi'^{\alpha+\beta}\rangle. \end{equation} 特别地(注意$\varphi_1=\langle~\varphi_1\rangle$), \begin{equation}\langle \varphi'^{k}_1\rangle \diamond \langle \varphi'^m_1\rangle =\langle \varphi'^{(k+m)}_1\rangle = \underbrace{\varphi_1\diamond \cdots \diamond \varphi_1}_{k+m\text{ 个}}=:\varphi_1^{\diamond (k+m)}. \end{equation} 另一方面, 对于任意$f\in~\mathcal{S}$,有 \begin{equation}X_f=\eta(f)=\eta \bigg (\sum_{n\geq 1} (f_n,f)_{L^2}\cdot f_n \bigg )=\sum_{n\geq 1}(f_n,f)_{L^2}\cdot X_{f_n}=\sum_{n\geq 1}(f_n,f)_{L^2}\cdot \varphi_n. \end{equation} 进而, 对$g_1,\ldots,~g_m\in~\mathcal{S}$,有 \begin{equation}\begin{aligned} \langle X_{g_1}, \ldots , X_{g_m}\rangle &= \bigg \langle \sum_{n\geq 1}(f_n,g_1)_{L^2}\cdot \varphi_n, \ldots , \sum_{n\geq 1}(f_n,g_m)_{L^2}\cdot \varphi_n \bigg \rangle \\ &= \bigg ( \sum_{n\geq 1}(f_n,g_1)_{L^2}\cdot \varphi_n\bigg )\diamond \cdots \diamond \bigg (\sum_{n\geq 1}(f_n,g_m)_{L^2}\cdot \varphi_n \bigg ) \\ &=X_{g_1}\diamond \cdots \diamond X_{g_m}. \end{aligned} \end{equation} 这说明, $(\mathcal{S}^{-1})$上的 Wick 乘法运算$\diamond$与 附录sect. 3.2 中的 Wick 积在某种意义上是一致的. 这同时意味着对于任意$g_1,\ldots,~g_m\in~\mathcal{S}$,有 \begin{equation} \langle \eta^{\otimes m}\rangle (g_1\otimes \cdots \otimes g_m)= (\eta(t_1,x_1)\diamond \cdots\diamond \eta(t_m,x_m) )(g_1\otimes \cdots \otimes g_m). \tag{71}\end{equation} 也就是说, 附录sect. 3.2 中定义的时空白噪声间的 Wick 积与用$\diamond$运算定义的时空白噪声之间的乘积在广义函数意义下是一致的.

Remark 18. 取上述的$f_1$及$\varphi_1=X_{f_1}$. 注意, \begin{equation}\langle X_{f_1}, X_{f_1}\rangle = X_{f_1}\diamond X_{f_1}=H_2(X_{f_1}), \end{equation} \begin{equation}\langle H_2(X_{f_1}), X_{f_1}\rangle \neq \langle X_{f_1}, X_{f_1}, X_{f_1}\rangle =X_{f_1}\diamond X_{f_1}\diamond X_{f_1}= H_2(X_{f_1})\diamond X_{f_1}. \end{equation} 这说明, 并非所有随机变量之间的 Wick 积都与$\diamond$运算一致.


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