Itô公式(参见文献[16-17])是随机分析和量子随机分析的基本结果. Itô随机积分已经被广泛应用于数学金融、随机控制和滤波等问题中(参见文献[1-2]).关于随机游动的Itô公式在文献[3-4]已被研究过.然而, 针对量子情形, Konno最近在文献[5]中研究了一维离散时间量子随机游动(简记为DQRW)的Itô公式.随后, Ampadu在平方整数格$Z_{2}$上将DQRW的Itô公式推广到二维情形(参见文献[6]).受其研究思想的启发, 基于文献[18]的研究成果, 作者再度考虑将关于CQRW的Itô公式开拓到$d$ -维情形.其中$d\in{\Bbb N}$且$d\geq 3, {\Bbb N}$是正整数集.

本文的写作目的是针对多维CQRW的Itô公式进一步展开研究.基于这一目标, 我们首要工作是建立高维CQRW模型.需要说明的是这方面的工作已在图上被研究过(参见文献[7-10, 13-14, 19]).但是还需强调这儿的延拓是一种自然的延拓, 是将单粒子游动延拓到多质点游动的情形.并且我们在上下文中也考虑了量子随机游动(简记为QRW)的纠缠与相互作用.

行文的内容安排如下:第二节主要简要回顾了高维CQRW的预备知识; 高维CQRW的Itô型公式将在第三节建立; 作为应用, 第四节给出了关于高维CQRW的Tanaka公式.

我们将在此小节简要回顾高维QRW的有关概念.在一个可供选择的Hilbert空间中QRW被看成是一个(连续时间) U -演化.假设$d\geq 1$是任意整数, $Z_{d}$是$d$ -维整数格.我们继续以$Z_{d}$上的QRW (参见文献[14-15])作为研究对象.因此其手征态在数学上可理解为Hilbert空间中的$2d$ -维向量, 并且通过$2d\times2d$ U -矩阵作用发生旋转.下面给出矩阵的变换.

对于$j=1, 2, \cdots, d$, 假设$A^{\pm}$是依如下方式定义的$2d\times2d$矩阵

$ A_{j}^{-}(l,\text{ }m):=\left\{ \begin{array}{*{35}{l}} 1,\text{ }若(l,\text{ }m)=(2j-1,\text{ }2j-1),\text{ } \\ 0,\text{ }其他 \\ \end{array} \right. $

和

$ A_{j}^{+}(l,m):=\left\{ \begin{array}{*{35}{l}} 1, & 若(l,m)=(2j,2j), \\ 0, & 其他. \\ \end{array} \right. $

(2.1)

由此可定义

$ U_{j}^{\pm }:=A_{j}^{\pm }U, ~~~~~~~j=1, 2, \cdots, d. $

(2.2)

其中

$ U=\left( \begin{matrix} {{l}_{1}}~~ & {{l}_{2}} \\ {{l}_{3}}~~ & {{l}_{4}} \\ \end{matrix} \right), $

这里$l_{i}\in{\Bbb C}, i=1, 2, 3, 4$. ${\Bbb C}$是复数集.

注意到:当$d=1$时, $U^{-}_{1}$和$U^{+}_{1}$各自分别对应于$L$和$R$, 具有如下形式

$ L=\left( \begin{matrix} 1 \\ 0 \\ \end{matrix} \right)~~~\text{ }和\text{ }~~~R=\left( \begin{matrix} 0 \\ 1 \\ \end{matrix} \right). $

(2.3)

从而QRW在$Z_{d}$上依据其手征态在每一时刻向左或向右移动一步, 向前或向后移动一步, 向上或向下移动一步, $\cdots$如此循环往复.因此, $d$ -维QRW可看成是Hilbert空间中的一种演化

$ {\cal H}:=l^{2}({\Bbb Z}_{d}, {\Bbb C}^{2d}) \approx\bigoplus_{x\in{\Bbb Z}_{d}}{\cal H}_{x}, $

(2.4)

其中${\cal H}_{x}$是${\Bbb Z}_{2d}$的复制, 或者等价于其Fourier变换空间

$ \hat{{\cal H}}:=L^{2}({\Bbb K}^{d}, {\Bbb C}^{2d}) \approx\int^{\oplus}_{{\Bbb K}^{d}}h_{k}{\rm d}k, $

(2.5)

这里$h_{k}$是${\Bbb C}^{2d}$的副本. $(2.4)$和$(2.5)$式之间的变换方式如下所示

$ \psi=(\psi(x))_{x\in{\Bbb Z}^{d}}\mapsto\hat{\psi}=(\hat{\psi}(k))_{k\in{\Bbb K}^{d}}, \\ (\hat{\psi}(k))=\sum_{x\in{\Bbb Z}^{d}}e_{x}(k)\psi(x), $

(2.6)

其中$e_{x}(k)=(2\Pi)^{-d/2}{\rm e}^{{\rm i}kx}$.依据文献[11-12]一维QRW具有如下演化方式

$ \psi_{n+1}(x)=L\psi_{n}(x+1)+R\psi_{n}(x-1), $

(2.7)

这儿$|\psi_{n}(x)\rangle$是手征态.现在我们着手将上述的演化进行延拓为

$ \psi_{n+1}(x)= \sum_{i=1}^{d}(U_{j}^{(-)}(T_{j}\psi_{n})(x)+U_{j}^{(+)}(T_{j}^{\ast}\psi_{n})(x)), ~~x\in{\Bbb Z}_{d}, $

(2.8)

其中$T_{j}, j=1, 2, \cdots, d$是将第$j$个坐标轴, 按照下面的方式进行变换

$ (T_{j}\psi)(x)=\psi(x+u_{j}), $

(2.9)

且$u_{j}$是第$j$个坐标轴上的单位向量.而$\psi_{n}$则是依据以下方式进行变换

$ \psi_{n}=(\tilde{L}T+\tilde{R}T^{\ast})^{n}\psi_{0}, ~~\psi_{0}\in {\cal H}, $

(2.10)

这儿的$T$意指空间$l^{2}({\Bbb Z})$上的左变换, 具有如下形式

$ (Ta)(x)=a(x+1), ~~~~\mbox{对任意}~~~ a =(a(x))_{x\in{\Bbb Z}}. $

(2.11)

并且$T$是$U$ -映射, 其共轭映射是具有以下特征的右变换

$ (T^{\ast}a)(x)=a(x-1), ~~~~\mbox{对任意}~~~ a =(a(x))_{x\in{\Bbb Z}}. $

(2.12)

这样算子$T$自然地可延拓到${\cal H}=\bigoplus\limits_{x\in{\Bbb Z}}{\cal H}_{x}$.因此方程$(2.8)$有解, 且形如

$ \psi_{n}=\left(\sum_{j=1}^{d} \widetilde{U_{j}^{(-)}}T_{j}+ \widetilde{U_{j}^{(+)}(T_{j}^{\ast}})\right)\psi_{0}, $

(2.13)

其中$\widetilde{U_{j}^{\pm}}$是将$U_{j}^{\pm}$延拓到空间${\cal H}$得到的结果.如果我们在‘Fourier’ -变换空间$\hat{{\cal H}}$中重新表示$(2.13)$式, 那么将有

$ \hat{\psi}_{n}(k)=(U(k))^{n}\hat{\psi}_{0}(k), ~~k\in{\Bbb K}, $

(2.14)

这儿$U(K)$是$U$ -矩阵, 表示形式如下

$ U(k) =\sum_{j=1}^{d}({\rm e}^{-{\rm i}k_{j}}A_{j}^{(-)}+{\rm e}^{{\rm i}k_{j}}A_{j}^{(+)})U =\left( \begin{array}{ccccc} {\rm e}^{-{\rm i}k_{1}} & & & & \\ & {\rm e}^{{\rm i}k_{1}} & & 0 & \\ & & \ddots & & \\ & 0 & & {\rm e}^{-{\rm i}k_{d}} & \\ & & & & {\rm e}^{{\rm i}k_{d}}\\ \end{array} \right)U. $

(2.15)

经过上述操作, 在一些适当技术条件下, QRW可以延拓到任意高维情形.假设$H(k)$是生成QRW的自伴随矩阵, 即它满足关系式

$ U(k)={\rm e}^{{\rm i}H(k)}, $

(2.16)

其中$H(k)$是自伴算子, 定义方式如下

$ H(k)=S(k-\theta_{1})\left( \begin{array}{cc} \gamma(k-\theta_{1}) & 0 \\ 0 & -\gamma(k-\theta_{1}) \\ \end{array} \right)S(k-\theta_{1})^{-1}, $

(2.17)

这儿$\gamma(k-\theta_{1})$可理解为周期为$2\Pi$的周期函数, 定义域为${\Bbb R}$且$\theta_{1}\in{\Bbb K}$.由于技术处理的需要, 我们做如下假设

$(\star)$$H(k)$中的特征值和特征向量在紧集${\Bbb K}^{d}$上都是连续可微的.

在行文中我们继续假定假设$(\star)$成立.

设$\gamma(k)$是定义在${\Bbb K}=(-\Pi, \Pi]$上的非负对称函数且$H(k)$可实现对角化, 形如

$ H(k)=S(k)\Gamma(k)S(k)^{\ast}, $

(2.18)

其中

$ \Gamma (k)=\left( \begin{matrix} \gamma _{1}^{(+)}(k) & {} & {} & {} & {} \\ {} & \gamma _{1}^{(-)}(k) & {} & 0 & {} \\ {} & {} & \ddots & {} & {} \\ {} & 0 & {} & \gamma _{d}^{(+)}(k) & {} \\ {} & {} & {} & {} & \gamma _{d}^{(-)}(k) \\ \end{matrix} \right) $

(2.19)

是特征值的对角矩阵, 而$S(k)$是$U$ -矩阵, 其列向量包含$H(k)$中的特征向量.

现在我们需要在‘Fourier‘变换空间中探究QRW的演化过程.易知变换算子$T$在‘Fourier‘变换空间中借助${\rm e}^{-{\rm i}k}$可被重写为一个乘算子.因此, $(2.10)$式中的演化在‘Fourier‘变换空间中具有如下表示形式

$ \hat\psi_{n}(k)=({\rm e}^{-{\rm i}k}L+{\rm e}^{{\rm i}k}R)\hat\psi_{0}(k) =\left( \begin{array}{cc} {\rm e}^{-{\rm i}k}l_{1}~~ & {\rm e}^{-{\rm i}k}l_{2} \\ {\rm e}^{{\rm i}k}l_{3}~~ & {\rm e}^{{\rm i}k}l_{4} \\ \end{array} \right)\hat\psi_{0}(k). $

(2.20)

注意到对每一个$k\in{\Bbb K}$, 矩阵

$ U(k)=\left( \begin{array}{cc} {\rm e}^{-{\rm i}k}l_{1}~~ & {\rm e}^{-{\rm i}k}l_{2} \\ {\rm e}^{{\rm i}k}l_{3} ~~& {\rm e}^{{\rm i}k}l_{4} \\ \end{array} \right) $

(2.21)

是${\Bbb C}^{2}$中的$U$ -矩阵, 因此$(2.20)$式中的演化在$\hat{H}$中理所当然也是$U$的.从而$(2.20)$式中的演化可被表示为

$ \hat\psi_{n}(k)=U(k)^{n}\hat\psi_{0}(k), $

这儿$U(k)$曾经出现在$(2.21)$式.借助文献[18]可知, $U$ -矩阵$U(k)$可被对角化为

$ U(k)=S(k-\theta_{1})\left( \begin{array}{cc} {\rm e}^{{\rm i}\gamma}(k-\theta_{1})~~ & 0 \\ 0 ~~& {\rm e}^{-{\rm i}\gamma}(k-\theta_{1}) \\ \end{array} \right)S(k-\theta_{1})^{-1}, $

(2.22)

因此我们可把它重写为形如$(2.16)$式.易见QRW的演化可被表示为

$ \hat{\psi_{n}}(k)={\rm e}^{{\rm i}nH(k)}\hat{\psi_{0}}(k), $

(2.23)

其中$H(k)$曾经出现在$(2.17)$式.此时可将DQRW延拓到CQRW.

定义2.1  设$H$是一个$2d\times 2d$ $U$ -矩阵.在‘Fourier’变换空间中${\Bbb Z}_{d}$上的CQRW通过$U$ -演化可被定义为

$ \hat{\psi_{t}}(k)={\rm e}^{{\rm i}tH(k)}\hat{\psi_{0}}(k), $

(2.24)

这儿$H(k)$是在$(2.17)$式中给出的自伴算子.

关于CQRW的简单图的直积问题, 作为一个例子, 我们可借助$d$ -维整数格${\Bbb Z}_{d}$ (是乘积图的一种简单结构)直接计算其波函数$|\psi_{t}\rangle$.

注2.1  相应的${\Bbb Z}_{d}$的正规化的邻接矩阵可看做是${\Bbb Z}^{2}$的直积

$ H=\frac{1}{n}\sum_{i=1}^{n}I_{2}\otimes\cdots\sigma_{x}\otimes\cdots\otimes I_{2}, $

(2.25)

其中和式中第$i$ -项具有形如‘Pauli’ -矩阵, $\sigma_{x}=\left( \begin{array}{cc} 0 ~~& 1 \\ 1 ~~& 0 \\ \end{array} \right)$出现在张量积中的第$i$ -位置.

因此有

$ \begin{array}[b]{rl} U_{t}&=\exp(-{\rm i}Ht)\\ &=I_{2}\otimes\cdots\otimes {\rm e}^{{\rm i}t\sigma_{x}/n}\otimes\cdots\otimes I_{2}\\ &={\rm e}^{{\rm i}t\sigma_{x}/n}\otimes\cdots\otimes {\rm e}^{{\rm i}t\sigma_{x}/n}\\ &= ({\rm e}^{{\rm i}t\sigma_{x}/n})^{\otimes n}\\ &=\left( \begin{array}{cc} \cos(t/n) ~~& {\rm i}\sin(t/n) \\ {\rm i}\sin(t/n)~~ &\cos(t/n) \\ \end{array} \right)^{\otimes n}, \end{array} $

(2.26)

这儿$B^{\otimes n}$是$B$的$n$个副本的张量积.若$|\psi_{0}\rangle = |0\rangle^{\otimes n}$, 则

$ |\psi_{t}=U_{t}|\psi_{0}\rangle=[\cos(t/n)+{\rm i}\sin(t/n)]^{\otimes n}. $

(2.27)

定义2.2  由此我们看到CQRW类似于$n$个无相互作用的量子比特系统.

此小节我们考虑在${\Bbb Z}_{d}$上建立$d$ -维CQRW ($d\geq 3$)的Itô公式, 其中${\Bbb Z}$是整数.设${\Bbb R}^{+}$是正实数且${\Bbb T}\in {\Bbb R}^{+}$.一个路径$w$可表示为$w=(w_{t_{0}}, w_{t_{1}}, \cdots, w_{t_{n}})$, 其中$w_{t_{i}}\in {\Bbb R}, t_{i}\in{\Bbb T}$, 且$|w_{t_{i}}-w_{t_{i-1}}|=1$.路径$w$的长度被定义为$|w|=n, n\in\{0, 1, \cdots, \}$.现在我们就可以考察路径空间$\Omega_{t_{n}}$上的CQRW.为此, 假定$w =(w_{t_{0}}(x_{0})=0, w_{t_{1}}(x_{1}), \cdots, w_{t_{n}}(x_{n}))\in\Omega_{t_{n}}$和$w_{t_{i}}(x_{i})=x_{t_{i}}$, 其意指游走者分别向左、向右、向下、向上、向前和向后等各个方向移动, 即指游走者在每一时间$t_{i}$步在位置$x_{i}$处的运动方向.例如, 当$i=1, 2, \cdots, d$时, 设$n, d\in{\Bbb N}=\{1, 2, \cdots, \}$.令

$ B_{t_{i}}=\{-t_{i}, -(t_{i}-1), \cdots, t_{i}-1, t_{i}\}, $

(3.1)

$ \Omega_{t_{i}}=B_{t_{i}}^{n+1}=\{-t_{i}, -(t_{i}-1), \cdots, t_{i}-1, t_{i}\}^{n+1}, $

(3.2)

$ w_{t_{i}}=(w_{t_{i}}(0)=0, w_{t_{i}}(1), w_{t_{i}}(2), \cdots, w_{t_{i}}(n))\in\Omega_{t_{i}}, $

(3.3)

$ v_{t_{i}}=(w_{t_{i}}(1), w_{t_{i}}(2)-w_{t_{i}}(1), \cdots, w_{t_{i}}(n)-w_{t_{i}}(n-1)), $

(3.4)

$ u_{t_{i}}=(I_{\{1\}}(v_{t_{i}}(1)), I_{\{1\}}(v_{t_{i}}(2)), \cdots, I_{\{1\}}(v_{t_{i}}(n))), $

(3.5)

其中$I_{A}(x)$是指集合$A$的示性函数.注意到$w_{t_{i}}(m_{j}+1)-w_{t_{i}}(m_{j})\in\{-1, 1\}$且$m_{j}\in{\Bbb N}, $ $j=1, 2, \cdots, d$.由此我们即可考察CQRW在下列空间上的行为特征, $\Omega_{t_{1}, t_{2}, \cdots, t_{d}}=B_{t_{1}}^{n+1}\times B_{t_{2}}^{n+1}\times\cdots\times B_{t_{d}}^{n+1}$.为此设

$ \begin{array}{rl} w_{t_{1}, t_{2}, \cdots, t_{d}}&=w_{t_{1}}\times w_{t_{2}}\times\cdots\times w_{t_{d}}\in\Omega_{t_{1}, t_{2}, \cdots, t_{d}}, \\ v_{t_{1}, t_{2}, \cdots, t_{d}}&=v_{t_{1}}\times v_{t_{2}}\times \cdots\times v_{t_{d}}, \\ u_{t_{1}, t_{2}, \cdots, t_{d}}&=u_{t_{1}}\times u_{t_{2}}\times \cdots\times u_{t_{d}}.\end{array} $

(3.6)

且有

$ {{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{m}_{1}}+1,{{m}_{2}},\cdots ,{{m}_{d}})-{{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}},\cdots ,{{m}_{d}})\in \{-1,1\} \\ \times \cdots \times \{-1,1\}={{\{-1,1\}}^{d}}, \\ {{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}}+1,\cdots ,{{m}_{d}})-{{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}},\cdots ,{{m}_{d}})\in \{-1,1\} \\ \times \{-1,1\}={{\{-1,1\}}^{d}}, \\ \cdots $

而且

$ \begin{array}{rl} &w_{t_{1}, t_{2}, \cdots, t_{d}}(m_{1}, m_{2}, \cdots, m_{d}+1)-w_{t_{1}, t_{2}, \cdots, t_{d}} (m_{1}, m_{2}, \cdots, m_{d})\in\{-1, 1\}\\ &\times\cdots\times\{-1, 1\}=\{-1, 1\}^{d}.\end{array} $

(3.7)

依据文献[18]的方法, 直接地计算有

命题3.1  对任意的$ m_{j}\in\{0, 1, 2, \cdots, n_{d}-1\}, t_{i}\in [0, {\Bbb T}], i, j=1, 2, \cdots, d$.设$f: {\Bbb Z}^{d}\rightarrow {\Bbb C}$.则有

(1)

$ f({{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{m}_{1}}+1,{{m}_{2}},\cdots ,{{m}_{d}}))-f({{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}},\cdots ,{{m}_{d}})) \\ =\frac{1}{2}\{f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})+1)-f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})-1)\} \\ \times ({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}}+1,\cdots ,{{m}_{d}})-{{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}}\cdots ,{{m}_{d}})) \\ +\frac{1}{2}\{f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})+1)-2f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})) \\ +f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})-1)\}, \\ f({{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}}+1,\cdots ,{{m}_{d}}))-f({{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}},\cdots ,{{m}_{d}}))\text{ } \\ =\frac{1}{2}\{f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})+1)-f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})-1)\} \\ \times ({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}}+1,\cdots ,{{m}_{d}})-{{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}}\cdots ,{{m}_{d}}))\text{ } \\ +\frac{1}{2}\{f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})+1)-2f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}}))\text{ } \\ +f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})-1)\}, \\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\cdots \\ f({{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}},\cdots ,{{m}_{d}}+1))-f({{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}},\cdots ,{{m}_{d}}))\text{ } \\ =\frac{1}{2}\{f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})+1)-f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})-1)\} \\ \times ({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}},\cdots ,{{m}_{d}}+1)-{{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}}\cdots ,{{m}_{d}}))\text{ } \\ +\frac{1}{2}\{f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})+1)-2f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}}))\text{ } \\ +f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})-1)\}. $

(2)

$ f({{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{n}_{1}},{{n}_{2}},\cdots ,{{n}_{d}}))-f({{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}(0,{{n}_{2}},\cdots ,{{n}_{d}}))\text{ } \\ =\frac{1}{2}\sum\limits_{{{m}_{j}}=0}^{{{n}_{d}}-1}{\{f(}{{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})+1)-f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})-1)\} \\ \times ({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}}+1,\cdots ,{{m}_{d}})-{{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}}\cdots ,{{m}_{d}}))\text{ } \\ +\frac{1}{2}\sum\limits_{{{m}_{j}}=0}^{{{n}_{d}}-1}{\{f(}{{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})+1)-2f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}}))\text{ } \\ +f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})-1)\},\text{ } \\ f({{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{n}_{1}},{{n}_{2}},\cdots ,{{n}_{d}}))-f({{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{n}_{1}},0,\cdots ,{{n}_{d}}))\text{ } \\ =\frac{1}{2}\sum\limits_{{{m}_{j}}=0}^{{{n}_{d}}-1}{\{f(}{{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})+1)-f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})-1)\} \\ \times ({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}}+1\cdots ,{{m}_{d}})-{{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}}\cdots ,{{m}_{d}})) \\ +\frac{1}{2}\sum\limits_{{{m}_{j}}=0}^{{{n}_{d}}-1}{\{f(}{{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})+1)-2f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})) \\ +f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})-1)\},\text{ } \\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\cdots \\ f({{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{n}_{1}},{{n}_{2}},\cdots ,{{n}_{d}}))-f({{w}_{{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{d}}}}({{n}_{1}},\cdots ,{{n}_{d-1}},0))\text{ } \\ =\frac{1}{2}\sum\limits_{{{m}_{j}}=0}^{{{n}_{d}}-1}{\{f(}{{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})+1)-f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})-1)\} \\ \times ({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}} +1)-{{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},{{m}_{2}}\cdots ,{{m}_{d}}))\text{ } \\ +\frac{1}{2}\sum\limits_{{{m}_{j}}=0}^{{{n}_{d}}-1}{\{f(}{{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})+1)-2f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})) \\ +f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}})-1)\}. $

证  式(2)容易证明.通过在式$(1)$的求和过程中关于指标$m_{1}, m_{2}, \cdots, m_{d}$分别从$0$一直加到$n_{d}-1$, 就可得到表达式的结果.特别地, 易知在式$(2)$的表达式中右边容易被遗漏, 依据求和符号选取适当的指标, 由$0$一直加到$n_{d}-1$即可得到结果.在式$(1)$中的求和过程中关于指标$m_{1}, m_{2}, \cdots, m_{d}$分别从$0$一直加到$n_{d}-1$, 和式左边变成一个可伸缩的总和, 即, 和式压缩后式$(2)$的表达式中左边仅仅为两项.接着来看式$(1)$的表达式是怎样的.例如, 设$w_{t_{1}, \cdots, t_{d}}(m_{1}+1, m_{2}, \cdots, m_{d})-w_{t_{1}, \cdots, t_{d}}(m_{1}, m_{2}, \cdots, m_{d})=1$, 则有

$ LHS=f(w_{t_{1}, \cdots, t_{d}}(m_{1}, \cdots, m_{d})+1)-f(w_{t_{1}, \cdots, t_{d}}(m_{1}, \cdots, m_{d})), $

由于

$ w_{t_{1}, \cdots, t_{d}}(m_{1}+1, m_{2}, \cdots, m_{d})-w_{t_{1}, \cdots, t_{d}}(m_{1}, m_{2}, \cdots, m_{d})=1, $

因此

$ w_{t_{1}, \cdots, t_{d}}(m_{1}+1, m_{2}, \cdots, m_{d})=w_{t_{1}, \cdots, t_{d}}(m_{1}, m_{2}, \cdots, m_{d})+1, $

从而有

$ \begin{eqnarray*} RHS&=&f(w_{t_{1}, \cdots, t_{d}}(m_{1}, \cdots, m_{d})+1)-f(w_{t_{1}, \cdots, t_{d}}(m_{1}, \cdots, m_{d}))\\ &=&f(w_{t_{1}, \cdots, t_{d}}(m_{1}+1, \cdots, m_{d}))-f(w_{t_{1}, \cdots, t_{d}}(m_{1}, \cdots, m_{d})). \end{eqnarray*} $

至此完成了证明.

注3.1  命题$3.1$的表达式中的第一部分$(1)$是高维CQRW在$d$ -维整数格${\Bbb Z}_{d}$上的Itô公式.

对任意$i=1, 2, \cdots, d$, 令

$ k_{i}=u_{t_{i}}(n_{i})2^{n_{i}-1}+u_{t_{i}}(n_{i}-1)2^{n_{i}-2}+\dots+u_{t_{i}}(2)2^{1} +u_{t_{i}}(1)2^{0}, $

(3.8)

且设

$ P_{w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}}=P_{w_{t_{1}}^{(k_{1})}}\otimes\cdots\otimes P_{w_{t_{d}}^{(k_{d})}}, $

(3.9)

其中

$ P_{w_{t_{i}}^{(k_{i})}}=P_{v_{t_{i}}^{(k_{i})}(n_{d})}\cdots P_{v_{t_{i}}^{(k_{i})}(2)} P_{v_{t_{i}}^{(k_{i})}(1)}. $

(3.10)

那么我们立即得到

定理3.1  对任意的$m_{j}\in\{0, 1, 2, \cdots, n_{d}-1\}, t_{i}\in [0, {\Bbb T}], i, j=1, 2, \cdots, d$, 设$f: {\Bbb Z}^{d}\rightarrow {\Bbb C}$.则有

(1)

$ \sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} \{f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1}+1,\cdots,m_{d}))-f(w_{t_{1}, \cdots,t_{d}}(m_{1}, \cdots,m_{d}))\}P_{w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}}\\ =\frac{1}{2}\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} \{f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d})+1) -f(w_{t_{1},\cdots,t_{d}}(m_{1},\cdots,m_{d})\\ -1)\}(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})} (m_{1}+1,\cdots,m_{d}) -w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d})) P_{w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}}\\ +\frac{1}{2}\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} \{f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d})+1)- 2f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d}))\\ +f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d})-1)\} P_{w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}}, \\ \sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} \{f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},m_{2}+1\cdots,m_{d}))- f(w_{t_{1},\cdots,t_{d}} (m_{1},\cdots,m_{d}))\}P_{w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}}\\ =\frac{1}{2}\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} \{f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d})+1) -f(w_{t_{1},\cdots,t_{d}}(m_{1},\cdots,m_{d})\\ -1)\}(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})} (m_{1}+1,\cdots,m_{d}) -w_{n_{1},\cdots,n_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d})) P_{w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}}\\ +\frac{1}{2}\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} \{f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d})+1)- 2f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d}))\\ +f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d})-1)\} P_{w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}}, \\ \cdots \\ \sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} \{f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},m_{2}\cdots,m_{d}+1)) -f(w_{t_{1},\cdots,t_{d}} (m_{1},\cdots,m_{d}))\}P_{w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}}\\ =\frac{1}{2}\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} \{f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d})+1) -f(w_{t_{1},\cdots,t_{d}}(m_{1},\cdots,m_{d})\\ -1)\}(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})} (m_{1}+1,\cdots,m_{d}) -w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d})) P_{w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}}\\+\frac{1}{2}\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} \{f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d})+1)- f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d}))\\ +f(w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}(m_{1},\cdots,m_{d})-1)\} P_{w_{t_{1},\cdots,t_{d}}^{(k_{1},\cdots,k_{d})}}. $

(2)

$ \sum\limits_{{{k}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }\sum\limits_{{{k}_{d}}=0}^{{{2}^{{{n}_{d}}}}-1}{\{f(}w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{n}_{1}},{{n}_{2}}\cdots ,{{n}_{d}}))-f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}(0,\cdots ,{{n}_{d}}))\}{{P}_{w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}}} \\ =\frac{1}{2}\sum\limits_{{{k}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }\sum\limits_{{{k}_{d}}=0}^{{{2}^{{{n}_{d}}}}-1}{\sum\limits_{{{m}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }}\underset{{{m}_{d}}=0}{\overset{{{2}^{{{n}_{d}}}}-1}{\mathop \sum }}\, \{f(w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}})+1)-f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}}) \\ -1)\}(w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}}+1,\cdots ,{{m}_{d}})-w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}})){{P}_{w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}}} \\ +\frac{1}{2}\sum\limits_{{{k}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }\sum\limits_{{{k}_{d}}=0}^{{{2}^{{{n}_{d}}}}-1}{\sum\limits_{{{m}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }}\sum\limits_{{{m}_{d}}=0}^{{{2}^{{{n}_{d}}}}-1}{\{f(}w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}})+1) \\ -2f(w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}}))+f(w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}})-1)\}{{P}_{w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}}}, \\ \sum\limits_{{{k}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }\sum\limits_{{{k}_{d}}=0}^{{{2}^{{{n}_{d}}}}-1}{\{f(}w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{n}_{1}},{{n}_{2}}\cdots ,{{n}_{d}}))-f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{n}_{1}},0,\cdots ,{{n}_{d}}))\}{{P}_{w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}}} \\ =\frac{1}{2}\sum\limits_{{{k}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }\sum\limits_{{{k}_{d}}=0}^{{{2}^{{{n}_{d}}}}-1}{\sum\limits_{{{m}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }}\sum\limits_{{{m}_{d}}=0}^{{{2}^{{{n}_{d}}}}-1}{\{f(}w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}})+1)-f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}}) \\ -1)\}(w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},{{m}_{2}}+1,\cdots ,{{m}_{d}}) -w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}})){{P}_{w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}}} \\ +\frac{1}{2}\sum\limits_{{{k}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }\sum\limits_{{{k}_{d}}=0}^{{{2}^{{{n}_{d}}}}-1}{\sum\limits_{{{m}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }}\sum\limits_{{{m}_{d}}=0}^{{{2}^{{{n}_{d}}}}-1}{\{f(}w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}})+1) \\ -2f(w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}}))+f(w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}})-1)\}{{P}_{w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}}},\text{ } \\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\cdots \\ \sum\limits_{{{k}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }\sum\limits_{{{k}_{d}}=0}^{{{2}^{{{n}_{d}}}}-1}{\{f(}w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{n}_{1}},{{n}_{2}}\cdots ,{{n}_{d}}))-f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{n}_{1}},\cdots ,{{n}_{d-1}},0))\}{{P}_{w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}}} \\ =\frac{1}{2}\sum\limits_{{{k}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }\underset{{{k}_{d}}=0}{\overset{{{2}^{{{n}_{d}}}}-1}{\mathop \sum }}\, \sum\limits_{{{m}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }\sum\limits_{{{m}_{d}}=0}^{{{2}^{{{n}_{d}}}}-1}{\{f(}w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}})+1)-f({{w}_{{{t}_{1}},\cdots ,{{t}_{d}}}}({{m}_{1}},\cdots ,{{m}_{d}}) \\ -1)\}(w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},{{m}_{2}},\cdots ,{{m}_{d}}+1) -w_{{{n}_{1}},\cdots ,{{n}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}})){{P}_{w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}}} \\ +\frac{1}{2}\sum\limits_{{{k}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }\sum\limits_{{{k}_{d}}=0}^{{{2}^{{{n}_{d}}}}-1}{\sum\limits_{{{m}_{1}}=0}^{{{2}^{{{n}_{d}}}}-1}{\cdots }}\sum\limits_{{{m}_{d}}=0}^{{{2}^{{{n}_{d}}}}-1}{\{f(}w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}})+1) \\ -2f(w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}}))+f(w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},\cdots ,{{m}_{d}})-1)\}{{P}_{w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}}}. $

注3.2  我们考虑:当$\sum\limits_{i=1}^{d}p_{i}=1$, 其中$p_{i}, i=1, 2, \cdots, d\in{\Bbb N}$对应于QRW在$d$ -维整数格${\Bbb Z}_{d}$上向左移动一步, 或向右、向下、向上、向前、向后, $\cdots$, 移动一步的概率.则有

$ P_{(\mp1, 0, \cdots, 0)}\rightarrow p_{1}\in[0, 1, ~~P_{(0, \mp1, \cdots, 0)}\rightarrow p_{2}\in[0, 1], \cdots, P_{(0, 0, \cdots, \mp1)}\rightarrow p_{d}\in[0, 1]. $

那么若$p_{1}=p_{2}=\cdots=p_{d}=\frac{1}{d}, d\in{\Bbb N}$时, 则定理$3.3$是对应于DQRW在非对称的${\Bbb Z}_{d}$上移动一步的概率.此结果对于简单对称随机游动在$d$ -维整数格${\Bbb Z}_{d}$上移动一步的概率也有相应的结果.

此小节依据文献[18]我们将借助高维CQRW的Itô公式建立相应的Tanaka公式.

按照定理$3.3$的表达式的第一部分$(1)$, 我们得到

$ f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}+1, \cdots, m_{d})) =|w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}+1, \cdots, m_{d})|, \\ f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}+1, \cdots, m_{d})) =|w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}+1, \cdots, m_{d})|, \\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\cdots \\ f(w_{{{t}_{1}}, \cdots, {{t}_{d}}}^{({{k}_{1}}, \cdots, {{k}_{d}})}({{m}_{1}}, {{m}_{2}}, \cdots, {{m}_{d}}+1))=|w_{{{t}_{1}}, \cdots, {{t}_{d}}}^{({{k}_{1}}, \cdots, {{k}_{d}})}({{m}_{1}}, {{m}_{2}}, \cdots, {{m}_{d}}+1)|. $

(4.1)

$ f(w_{{{t}_{1}},\cdots ,{{t}_{d}}}^{({{k}_{1}},\cdots ,{{k}_{d}})}({{m}_{1}},{{m}_{2}},\cdots ,{{m}_{d}}))=0. $

(4.2)

对任意的$k_{i}, i=1, 2, \cdots, d, $有下式成立

$ \begin{eqnarray} &&\frac{f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}, \cdots, m_{d})+1) -f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}, \cdots, m_{d})-1))}{2} \nonumber\\ &&={\rm sgn}(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}, \cdots, m_{d})), \end{eqnarray} $

(4.3)

其中sgn$(\cdot)$意指符号函数.

$ \begin{eqnarray} &&\frac{f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, \cdots, m_{d})+1) -2f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, \cdots, m_{d})) +f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, \cdots, m_{d})-1)}{2}\nonumber\\ &&=I_{\{0\}}(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}, \cdots, m_{d})). \end{eqnarray} $

(4.4)

对定理$3.3$的第一部分$(1)$中的第二个表达式采用类似的替换, 我们即可获得$d$ -维CQRW在整数格${\Bbb Z}^{d}$上的Tanaka公式如下.

推论4.1  假设题设条件与定理$3.3$一样.则有

$ \begin{eqnarray} &&\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} |w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}+1, \cdots, m_{d})| P_{w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}}\nonumber\\ &=&\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} {\rm sgn}(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, \cdots, m_{d})) (w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}+1, \cdots, m_{d}) \nonumber\\ &&-(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, \cdots, m_{d})) ) P_{w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}} \nonumber\\ &&+\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1}I_{\{0\}} (w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, \cdots, m_{d})) P_{w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}}, \nonumber\\ &&\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} |w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}+1, \cdots, m_{d})| P_{w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}}\nonumber\\ &=&\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} {\rm sgn}(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, \cdots, m_{d})) (w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}+1, \cdots, m_{d})\nonumber\\ &&-(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, \cdots, m_{d})) ) P_{w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}}\nonumber\\ &&+\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1}I_{\{0\}} (w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, \cdots, m_{d})) P_{w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}}, \nonumber\\ &&\cdots \nonumber\\ &&\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} |w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}, \cdots, m_{d}+1)| P_{w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}} \nonumber\\ &=&\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1} {\rm sgn}(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, \cdots, m_{d})) (w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}, \cdots, m_{d}+1) \nonumber\\ &&-(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, \cdots, m_{d})) ) P_{w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}} \nonumber\\ &&+\sum_{k_{1}=0}^{2^{n_{d}}-1}\cdots\sum_{k_{d}=0}^{2^{n_{d}}-1}I_{\{0\}} (w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, \cdots, m_{d})) P_{w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}}. \end{eqnarray} $

(4.5)

依据文献[5-6]和[18]的思想, 对任意的$d\in{\Bbb N}, d\geq2$, 有函数$f(x_{1}, x_{2}, \cdots, x_{d})$.而要弄清$d$个变量的函数之间的关系将是一个十分有趣的问题.众所周知$\vartriangle f=f_{x_{1}}\vartriangle x_{1}+f_{x_{2}}\vartriangle x_{2}+\cdots+f_{x_{d}}\vartriangle x_{d}$, 利用已有关系式, 我们将得到另外一个$d$ -维CQRW在整数格${\Bbb Z}^{d}$上的Itô公式.这里仅仅把它看做一个猜想, 有朝一日验证或否定它都将是十分有趣的问题.

猜想4.1  设$f:{\Bbb Z}^{d}\rightarrow {\Bbb C}$.对任意的$m_{j}\in\{0, 1, 2, \cdots, n_{d}-1\}, t_{i}\in [0, {\Bbb T}], i, j=1, 2, \cdots, d$.有

$ \begin{eqnarray} &&f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}+1, \cdots, m_{d}+1))- f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, \cdots, m_{d})) \nonumber\\ &=&\frac{1}{2d}\{f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}+1, m_{2}, \cdots, m_{d})+1) -f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}-1, m_{2}, \cdots, m_{d}) \nonumber\\ &&-1)\}(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}+1, m_{2}, \cdots, m_{d}) -w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}, \cdots, m_{d})) \nonumber\\ &&+\frac{1}{2d}\{f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}+1, \cdots, m_{d})+1) -f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}-1, \cdots, m_{d}) \nonumber\\ &&-1)\}(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}+1, \cdots, m_{d}) -w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}, \cdots, m_{d})) \nonumber\\ &&+\cdots \nonumber\\ &&+\frac{1}{2d}\{f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}, \cdots, m_{d}+1)+1) -f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, \cdots, m_{d}-1) \nonumber\\ &&-1)\}(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}, \cdots, m_{d}+1) -w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}, \cdots, m_{d})) \nonumber\\ &&+\frac{1}{2d}\{f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}+1, m_{2}, \cdots, m_{d})+1) +f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}-1, m_{2}, \cdots, m_{d}) \nonumber\\ &&-1)+f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}+1, \cdots, m_{d})+1) +f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}-1, \cdots, m_{d})-1) \nonumber\\ &&+\cdots \nonumber\\ &&+f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}, \cdots, m_{d}+1)+1) +f(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}, \cdots, m_{d}-1) \nonumber\\ &&-1)-2df(w_{t_{1}, \cdots, t_{d}}^{(k_{1}, \cdots, k_{d})}(m_{1}, m_{2}, \cdots, m_{d}))\}. \end{eqnarray} $

(4.6)