作为具有自相似性和长相依性的分数布朗运动近年来备受人们的关注, 它在很多带有随机性的模型中可以作为一个合适的扰动项.然而在对分数布朗运动充分研究的基础上, 一些著名学者建议使用更一般的自相似随机过程作为随机模型的扰动项.作为分数布朗运动的推广, Houdré-Villa^[1]首次给出了双分数布朗运动的概念并研究了它的一些性质.双分数布朗运动$B^{H, K}=\{B_t^{H, K}, t\geq0\}$, $H\in(0, 1)$, $K\in(0, 1]$是一个中心的自相似的高斯过程, 其协方差函数为

$\mathbb{E}[B_t^{H, K}B_s^{H, K}]=\frac{1}{2^K}\big[\big(|t|^{2H}+|s|^{2H}\big)^K-|t-s|^{2HK}\big],\;s, t\geq0.$

(1.1)

记$B_0^{H, K}=0$.注意到, 当$K=1$时, 双分数布朗运动是分数布朗运动, 它具有分数布朗运动的一些性质(例如, 自相似性和Hölder连续性), 然而它的增量不是平稳的.关于双分数布朗运动的研究可参见Chen^[2], Luan^[3], Tuder-Xiao^[4], Wang^[5]等文献.

由于Varadhan^[6]的工作, 关于随机过程的局部时问题的研究已经变成一个重要的研究课题, Jiang-Wang^[7], Wu-Xiao^[8]分别研究了分数布朗运动的相遇局部时与相交局部时, Jiang-Wang^[9], Yan等^[10], Shen-Yan^[11]等分别研究了两个相互独立的一维双分数布朗运动的相遇局部时的光滑性, Xiao^[12]和Wu-Xiao^[13]研究了各向异性的高斯随机场上的局部时.受上述文献的启发, 本文主要研究了两个相互独立的多参数双分数布朗运动的相遇局部时的存在性与联合连续性.下面介绍多参数双分数布朗运动的定义.

任意给定向量$H=(H_1, \cdots, H_N)\in(0, 1)^N$, $K=(K_1, \cdots, K_N)\in(0, 1]^N$, 如果实值中心高斯随机场$B_0^{H, K}=\{B_0^{H, K}(t), t\in\mathbb{R}_+^N\}$的协方差函数为

$\mathbb{E}\bigg(B_0^{H, K}(t)B_0^{H, K}(s)\bigg)=\prod\limits_{j=1}^N\frac{1}{2^{K_j}}\bigg[\bigg(s_j^{2H_j}+t_j^{2H_j}\bigg)^{K_j} -|t_j-s_j|^{2H_jK_j}\bigg], \quad s, t\in\mathbb{R}_+^N,$

(1.2)

则称$B_0^{H, K}$是指数为$H, K$的$(N, 1)$双分数布朗运动.在(1.2) 中$B_0^{H, K}$是各向异性的高斯随机场, 且对任意的$t\in\partial\mathbb{R}_+^N$, 有$B_0^{H, K}(0)=0$, 这里的$\partial\mathbb{R}_+^N$是$\mathbb{R}_+^N$的边界.

设$B_1^{H, K}, \cdots, B_d^{H, K}$是相互独立的, 且与$B_0^{H, K}$同分布, 则$B^{H, K}=\{B^{H, K}(t), t\in\mathbb{R}_+^N\}$是取值于$\mathbb{R}^d$上的高斯随机场, 且

$B^{H, K}(t)=\big(B_1^{H, K}(t), \cdots, B_d^{H, K}(t)\big),\quad\forall t\in\mathbb{R}_+^N,$

(1.3)

此时称$B^{H, K}$以指标为$H, K$的$(N, d)-$双分数布朗运动, 且$B^{H, K}$是算子自相似过程.即对任意的常数$c>0$,

$\big\{B^{H, K}(c^A t)\quad t\in\mathbb{R}^N\big\}\stackrel{d}{=}\big\{c^N B^{H, K}(t)\quad t\in\mathbb{R}^N\big\},$

(1.4)

其中$A=(a_{ij})$是$N\times N$对角矩阵, 且$a_{ii}=\frac{1}{H_iK_i}, 1\leq i\leq N$.当$i\neq j$时, $a_{ij}=0$, $X\stackrel{d}{=}Y$表示两过程具有相同的有限维分布.当$N=1$, $d=1$时, $B^{H, K}$是指标为$H, K$的一维双分数布朗运动.

本文中的$\langle\cdot, \cdot\rangle$和$|\cdot|$表示在欧氏空间$\mathbb{R}^p$上的内积和范数.任意的正整数$p$, $\lambda_p$表示$\mathbb{R}^p$上的Lebesgue测度, $c_{i, j}>0$为有限的正常数.

为了本文研究的需要, 本节主要介绍并给出一些引理.

令$I=[\varepsilon, 1]^N, 0<\varepsilon<1$, 则$I$是$\mathbb{R}_+^N$的闭子区间.定义随机过程

$X_0(t)=B_0^{H_1, K_1}(t)-B_0^{H_2, K_2}(t)\quad t\in I,$

(2.1)

其中$H_i=(H_{i, 1}, \cdots, H_{i, N}), K_i=(K_{i, 1}, \cdots, K_{i, N}), i=\{1, 2\}$, 且记

$\sigma^2(t, s)=\mathbb{E}\big[X_0(t)-X_0(s)\big]^2\quad t, s\in I.$

(2.2)

引理2.1 ^[4] 对于任意的$\varepsilon>0$, $H=(H_1, \cdots, H_N)\in(0, 1)^N$和$K=(K_1, \cdots, K_N)\in(0, 1]^N$, 则存在有限的正常数$c_{2, 1}$和$c_{2, 2}$, 对任意的$s, t\in I$, 有

$c_{2, 1}\sum\limits_\limits{\ell=1}^N|s_\ell-t_\ell|^{2H_\ell K_\ell}\leq\mathbb{E}\bigg[\bigg(B_0^{H, K}(s)-B_0^{H, K}(t)\bigg)^2\bigg]\leq c_{2, 2}\sum\limits_\limits{\ell=1}^N|s_\ell-t_\ell|^{2H_\ell K_\ell},$

(2.3)

和

$c_{2, 1}\sum\limits_\limits{\ell=1}^N|s_\ell-t_\ell|^{2H_\ell K_\ell}\leq\det\textrm{Cov}\bigg(B_0^{H, K}(s), B_0^{H, K}(t)\bigg)\leq c_{2, 2}\sum\limits_\limits{\ell=1}^N|s_\ell-t_\ell|^{2H_\ell K_\ell}.$

(2.4)

引理2.2 ^[4] 实值双分数布朗运动$B_0^{H, K}=\big\{B_0^{H, K}(t), t\in\mathbb{R}_+^N\big\}$在$I=[\varepsilon, 1]^N$上满足强局部$\varphi$ -非确定性的, 即存在仅与$H, K, N, \varepsilon$有关的有限正常数$c_{2, 3}$, 使得对任意整数$n>1$和$u, t^1, \cdots, t^n\in I$, 有下面的式子成立

$\textrm{Var}\big(B_0^{H, K}(u)|B_0^{H, K}(t^1), \cdots, B_0^{H, K}(t^n)\big)\geq c_{2, 3}\sum\limits_{\ell=1}^N\min\limits_{0\leq k\leq n}|u_{\ell}-t_{\ell}^k|^{2H_{\ell}K_{\ell}},$

(2.5)

其中$t^0=0$.

引理2.3  令$X_0=\{X_0(t), t\in\mathbb{R^N}\}$是指标为$H_1, K_1, H_2, K_2$的实值高斯随机场(见公式(2.1)), 则它具有下面的性质:

(ⅰ)存在仅与$H_1, K_1, H_2, K_2, N, \varepsilon$有关的正常数$c_{2, 4}$和$c_{2, 5}$, 使得对任意$s, t\in I$有

$c_{2.4}\sum\limits_{\ell=1}^N|t_{\ell}-s_{\ell}|^{2H_{\ell}K_{\ell}}\leq\sigma^2(t, s)\leq c_{2.5}\sum\limits_{\ell=1}^N|t_{\ell}-s_{\ell}|^{2H_{\ell}K_{\ell}},$

(2.6)

其中$H_{\ell}K_{\ell}=\min\{H_{1, \ell} K_{1, \ell}, H_{2, \ell}K_{2, \ell}\}, \ell=1, \cdots, N$.在本文中, 假定$H_1K_1\leq H_2K_2\leq\cdots\leq H_NK_N$.

(ⅱ)存在一仅与$H_1, K_1, H_2, K_2, N, \varepsilon$有关的有限正常数$c_{2, 6}$, 对任意的整数$n>1$和$u, t^1, \cdots, t^n\in I$, 有下面的式子成立

$\textrm{Var}\big(X_0^{H, K}(u)|X_0^{H, K}(t^1), \cdots, X_0^{H, K}(t^n)\big)\geq c_{2, 6}\sum\limits_{\ell=1}^N\min\limits_{0\leq k\leq n} |u_{\ell}-t_{\ell}^k|^{2H_{\ell}K_{\ell}},$

(2.7)

其中$t^0=0$.

证 (ⅰ)由于$B_0^{H_1, K_1}$和$B_0^{H_2, K_2}$是相互独立的, 则对任意的$s, t\in I$

$\begin{aligned} \sigma^2(t, s)&=\mathrm{E}\big[\big(X_0(t)-X_0(s)\big)^2\big]\\ &=\mathrm{E}\big[B_0^{H_1, K_1}(t)-B_0^{H_2, K_2}(t)-B_0^{H_1, K_1}(s)+B_0^{H_2, K_2}(s)\big]^2\\ &=\mathrm{E}\big[B_0^{H_1, K_1}(t)-B_0^{H_2, K_2}(s)\big]^2+\mathrm{E}\big[B_0^{H_1, K_1}(t)-B_0^{H_2, K_2}(s)\big]^2. \end{aligned}$

(2.8)

根据引理2.1, 存在依赖于$H_1, K_1, H_2, K_2, N, \varepsilon$有限正常数$c_{2, 7}$和$c_{2, 8}$, 使得

$c_{2.7}\bigg(\sum\limits_\limits{i=1}^2\sum\limits_\limits{\ell=1}^N|s_\ell-t_\ell|^{2H_{i, \ell}K_{i, \ell}}\bigg) \leq\sigma^2(t, s)\leq c_{2.8}\bigg(\sum\limits_\limits{i=1}^2\sum\limits_\limits{\ell=1}^N|s_\ell-t_\ell|^{2H_{i, \ell}K_{i, \ell}}\bigg).$

(2.9)

记$H_{\ell}K_{\ell}=\min\{H_{1, \ell} K_{1, \ell}, H_{2, \ell}K_{2, \ell}\}, \ell=1, \cdots, N$, 可得

$\sum\limits_{\ell=1}^N|t_\ell-s_\ell|^{2H_\ell K_\ell}\leq\sum\limits_\limits{\ell=1}^N|t_\ell-s_\ell|^{2H_{1, \ell}K_{1, \ell}}+ \sum\limits_{\ell=1}^N|t_\ell-s_\ell|^{2H_{2, \ell}K_{2, \ell}}\leq2\sum\limits_{\ell=1}^N|t_\ell-s_\ell|^{2H_\ell K_\ell}.$

(2.10)

(ⅱ)由于$B_0^{H_1, K_1}$和$B_0^{H_2, K_2}$是相互独立的以及条件方差定义可知

$\begin{aligned} &\textrm{Var}\big(X_0(u)|X_0(t^1), \cdots, X_0(t^n)) =\inf\limits_{a_i\in\mathbb{R}, 1\leq i\leq n}\mathbb{E}\bigg[\bigg(x_0(u))-\sum\limits_{i=1}^{n}a_iX_0(t^i)\bigg)^2\bigg]\\ =&\inf\limits_{a_i\in \mathbb{R}, 1\leq i\leq n}\bigg\{\mathbb{E}\bigg[\bigg(B_0^{H_1, K_1}(u)-\sum\limits_{i=1}^{n}a_iB_0^{H_1, K_1}(t^i)\bigg)^2\bigg]\\ &+\mathbb{E}\bigg[\bigg(B_0^{H_2, K_2}(u)-\sum\limits_{i=1}^{n}a_iB_0^{H_2, K_2}(t^i)\bigg)^2\bigg]\bigg\}\\ \geq&\inf\limits_{a_i\in \mathbb{R}, 1\leq i\leq n}\mathbb{E}\bigg[\bigg(B_0^{H_1, K_1}(u)-\sum\limits_{i=1}^{n}a_iB_0^{H_1, K_1}(t^i)\bigg)^2\bigg]\\ &+\inf\limits_{b_i\in \mathbb{R}, 1\leq i\leq n}\mathbb{E}\bigg[\bigg(B_0^{H_2, K_2}(u)-\sum\limits_{i=1}^{n}b_iB_0^{H_2, K_2}(t^i)\bigg)^2\bigg]\\ =&\textrm{Var}\big(B_0^{H_1, K_1}(u)|B_0^{H_1, K_1}(t^1), \cdots, B_0^{H_1, K_1}(t^n)\big)\\ &+\textrm{Var}\big(B_0^{H_2, K_2}(u)|B_0^{H_2, K_2}(t^1), \cdots, B_0^{H_2, K_2}(t^n)\big)\\ \geq& c_{2.3}\bigg(\sum\limits_\limits{\ell=1}^N\min\limits_{0\leq k\leq n}{|u_{\ell}-t_\ell^k|}^{2H_{1, \ell}K_{1, \ell}}+\sum\limits_\limits{\ell=1}^N\min\limits_{0\leq k\leq n}{|u_\ell-t_\ell^k|}^{2H_{2, \ell}K_{2, \ell}}\bigg). \end{aligned}$

(2.11)

由引理2.2和(2.10) 式可得性质(ⅱ).即证明了(2.1) 式中的高斯随机场$X_0(t)$分别满足Xiao^[12]中的条件(C1) 和(C3).

结合引理2.3和

$\det\textrm{Cov}(Z_1, \cdots, Z_n)=\textrm{Var}(Z_1)\prod\limits_{k=2}^{n}\textrm{Var}(Z_k|Z_1, \cdots, Z_{k-1}),$

(2.12)

其中$(Z_1, \cdots, Z_n)$为任意的高斯随机向量.则对任意的$t^1, \cdots, t^n\in\mathbb{R}^N$有

$\begin{aligned} &\det\textrm{Cov}\big(X_0(t^1), \cdots, X_0(t^n)\big)\\ \geq&\prod\limits_{j=1}^{n}\big[\textrm{Var}\big(B_0^{H_1, K_1}(t^j)|B_0^{H_1, K_1}(t^1), \cdots, B_0^{H_1, K_1}(t^{j-1})\big)\\&+ \textrm{Var}\big(B_0^{H_2, K_2}(t^j)|B_0^{H_2, K_2}(t^1), \cdots, B_0^{H_2, K_2}(t^{j-1})\big)\big]\\ \geq& c_{2.9}^n\prod\limits_{j=1}^{n}\bigg(\sum\limits_\limits{\ell=1}^N\min\limits_{0\leq k\leq{j-1}}{|t_\ell^j-t_\ell^k|}^{2H_{1, \ell}K_{1, \ell}}+\sum\limits_\limits{\ell=1}^N\min\limits_{0\leq k\leq{j-1}}{|t_\ell^j-t_\ell^k|}^{2H_{2, \ell}K_{2, \ell}}\bigg)\\ \geq& c_{2.10}\prod\limits_{j=1}^n\bigg(\sum\limits_\limits{\ell=1}^N\min\limits_{0\leq k\leq{j-1}}{|t_\ell^j-t_\ell^k|}^{2H_\ell K_\ell}\bigg). \end{aligned}$

(2.13)

引理2.4 ^[8] 设$\beta$, $\gamma$, 和$p$都是有限正常数, 则对任意$A \in (0, 1), $

$\int_0^1\frac{r^{p-1}}{(A+r^{\gamma})^{\beta}}\mathrm{d}r\asymp \begin {cases} A^{\frac{p}{\gamma}-\beta}, \quad &若 \quad{\beta\gamma>p}, \\ \log{\big(}1+A^{-1/\gamma}{\big)}, \quad &若 \quad{\beta\gamma=p}, \\ 1, \quad &若\quad{\beta\gamma<p}, \end {cases}$

(2.14)

其中$f(x)\asymp g(x)$表示$c\leq\frac{f(x)}{g(x)}\leq C$, $c, C$的值不依赖于$x\in(0, 1)$.

引理2.5 ^[14] 给定常数$a>0$和$0<\underline{b}<\bar{b}<1$, 则存在正常数$c_{2.9}$, 使得对任意的整数$n\geq1$和实数$0<r\leq1$, $b_j\in [\underline{b}, \bar{b}]$, 以及任意$s_0\in[0, a/2]$有

$\int_{a\leq s_1\leq\cdots\leq s_n\leq a+r}\prod\limits_{j=1}^n(s_j-s_{j-1})^{-b_j}\mathrm{d}s_1\cdots\mathrm{d}s_n \leq c_{2.11}(n!)^{(1/n)\sum\limits_{j=1}^nb_j-1}r^{n-\sum\limits_{j=2}^nb_j}.$

(2.15)

特别的, 当$b_j=\alpha\in[\underline{b}, \bar{b}]$, $j=1, \cdots, n$时有

$\int_{a\leq s_1\leq\cdots\leq s_n\leq a+r}\prod\limits_{j=1}^n(s_j-s_{j-1})^{-\alpha}\mathrm{d}s_1\cdots\mathrm{d}s_n \leq c_{2.12}(n!)^{\alpha-1}r^{n(1-(1-\frac{1}{n})\alpha)}.$

(2.16)

引理2.6 ^[8] 设$Z_1, \cdots, Z_n$是零均值的高斯随机变量, 并且是线性无关的, 则对任意非负的Borel函数$g:\mathbb{R}\rightarrow\mathbb{R}_+$, 有

$\quad \int_{\mathbb{R}^n}{\exp}\bigg[-\frac{1}{2}{\hbox{Var}}\bigg(\sum\limits_{i=1}^{n}\nu_jZ_j\bigg)\bigg]d\nu_1\cdots\nu_n\nonumber\\ =\frac{(2\pi)^{(n-1)/2}}{({\hbox{detCov}}(Z_1, \cdots, Z_n))^{1/2}} {\int_{-\infty}^\infty}g\bigg(\frac{\nu}{\sigma_1}\bigg)e^{-\nu^2/2}d\nu,$

(2.17)

在这里${\sigma_1^2}={\textrm{Var}(Z_1|Z_2, \cdots, Z_n)}$是$Z_1$关于$Z_2, \cdots, Z_n$的条件方差.

引理2.7 ^[14] 对任意的$q\in[0, \sum\limits_{\ell=1}^N\frac{1}{H_{\ell}K_{\ell}}]$, 设$\tau\in\{1, \cdots, N\}$使得

$\sum\limits_\limits{\ell=1}^{\tau-1}\frac{1}{H_{\ell}K_{\ell}}\leq q\leq\sum\limits_\limits{\ell=1}^{\tau}\frac{1}{H_{\ell}K_{\ell}}.$

(2.18)

约定$\sum\limits_{\ell=1}^{0}\frac{1}{H_{\ell}K_{\ell}}:=0$.则存在只依赖于$H, K$的正常数$\delta_\tau\leq1$使得每一个$\delta\in(0, \delta_\tau)$, 可以找到关于$\tau$的实数$p_\ell\geq1 (1\leq\ell\leq\tau)$满足下列性质:

$\sum\limits_\limits{\ell=1}^{\tau}\frac{1}{p_\ell}=1, \frac{H_{\ell}K_{\ell}q}{p_\ell}<1, \forall\ell=1, \cdots, \tau$

(2.19)

和

$(1-\delta)\sum\limits_\limits{\ell=1}^{\tau}\frac{H_{\ell}K_{\ell}q}{p_\ell}\leq H_{\tau}K_{\tau}q+\tau-\sum\limits_\limits{\ell=1}^{\tau}\frac{H_{\tau}K_{\tau}}{H_{\ell}K_{\ell}}.$

(2.20)

此外, 若记$\alpha_\tau:=\sum\limits_{\ell=1}^{\tau}\frac{1}{H_{\ell}K_{\ell}}-q>0$, 对任意的实数$\rho\in(0, \frac{\alpha_\tau}{2\tau})$, 则存在$\ell_0\in\{1, \cdots, \tau\}$使得

$\frac{H_{\ell_0}K_{\ell_0}q}{p_{\ell_0}}+2H_{\ell_0}K_{\ell_0}\rho<1.$

(2.21)

设$Y(t)$是定义在$\mathbb{R}^p$上取值于$\mathbb{R}^q$的一任意Borel向量.对任意的Borel集$\mathrm{E}\subseteq\mathbb{R}^p$, 可以定义$Y$在$\mathrm{E}$上的占有测度为

$\mu_E(\bullet)=\lambda_p\{t\in E:Y(t)\in \bullet\}, $

其中$\lambda_p$是$\mathbb{R}^p$上的Lebesgue测度.若$\mu_E$关于Lebesgue测度$\lambda_p$是绝对连续的, 则称$Y(t)$在$\mathrm{E}$上的局部时存在, 记作$L(x, E)$, 它是$\mu_E$关于$\lambda_q$的Radon-Nikodym导数,

$L(x, E)=\frac{d\mu_E}{d\lambda_q}(x), \quad \forall x\in \mathbb{R}^q.$

上式中的$x$是空间变量, $\mathrm{E}$是时间变量.注意到, 若$Y(t)$在$\mathrm{E}$上有局部时, 则对任意的Borel集$\mathrm{S}\subset\mathrm{E}$, 则$L(x, S)$存在.

根据定理6.4 ^[15], 局部时是可测的且满足占有密度公式:对任意的Borel集$\mathrm{E}\subseteq\mathbb{R}^p$和所有可测函数$f:\mathbb{R}^q\longrightarrow\mathbb{R}_+$, 有

$\int_E f(Y(t))dt=\int_{\mathbb{R}^q}f(x)L(x, E)dx.$

假设矩形集$E=[a, a+h]\subset\mathbb{R}^p$, 其中$a\in\mathbb{R}^p$和$h\in\mathbb{R}_+^p$.如果局部时$L(x, [a, a+t])$在$(x, t)\in\mathbb{R}^q\times[0, h]$处是连续的函数, 则称$Y(t)$在$\mathrm{E}$上有联合连续的局部时.当局部时是联合连续时, $L(x, \cdot)$可以拓展到水平集上的有限的Borel测度, 即

$Y_E^{-1}(x)=\{t\in E : Y(t)=x\}, $

这使得局部时成为研究$Y$分形的有效工具.

根据(25.5), (25.7) ^[15]式, 对任意的$x, y\in\mathbb{R}^q$, 闭区间$E\subseteq\mathbb{R}^p$, 则对任意的整数$n\geq1$, 有

$\begin{aligned} \mathbb{E}[L(x, E)^n]=&(2\pi)^{-nq}\int_{E^n}\int_{\mathbb{R}^{nq}}\exp\bigg(-i\sum\limits_{j=1}^{n}\langle u_j, x\rangle\bigg)\\ &{\times}\mathbb{E}\exp\bigg(i\sum\limits_{j=1}^{n}\langle u_j, Y(t_j)\rangle\bigg)\mathrm{d}\bar{u}\mathrm{d}\bar{t} \end{aligned}$

(3.1)

和对任意的偶数$n\geq2$, 有

$\begin{aligned} \mathbb{E}\big[\big(L(x, E)-L(y, E)\big)^n\big]=&(2\pi)^{-nq}\int_{E^n}\int_{\mathbb{R}^{nq}}\prod\limits_{j=1}^{n}\big[e^{-i\langle u_j, x\rangle}-e^{-i\langle u_j, y\rangle}\big]\\ &{\times}\mathbb{E}\exp\bigg(i\sum\limits_{j=1}^{n}\langle u_j, Y(t_j)\rangle\bigg)\mathrm{d}\bar{u}\mathrm{d}\bar{t}, \end{aligned}$

(3.2)

其中$\bar{u}=(u_1, \cdots, u_n)$, $\bar{t}=(t_1, \cdots, t_n)$, 且${u_j}\in \mathbb{R}^q, {t_j}\in E$, $u_j=(u_{j, 1}, \cdots, u_{j, q})$.

设$\mathcal{A}$是包含所有闭区间的集合, 集合$I:=[s, t]^N\subseteq[\varepsilon, 1]^N$

定理3.1  设$B^{H_1, K_1}=\{B^{H_1, K_1}(t), t\in\mathbb{R}^N\}$和$B^{H_2, K_2}=\{B^{H_2, K_2}(t), t\in\mathbb{R}^N\}$是指数分别为$H_1, K_1$和$H_2, K_2$两个相互独立的$(N, d)$双分数布朗运动.设$X=\{X(t), t\in\mathbb{R}^N \}$是$(N, d)$ -高斯随机场, $X(t)=B^{H_1, K_1}(t)-B^{H_2, K_2}(t)$.对任意给定的集合$I\in\mathcal{A}$, $X$有局部时$L(x, I)\in L^2(\mathbb{P}\times\lambda_d)$当且仅当$\sum\limits_{\ell=1}^N\frac{1}{H_{\ell}K_{\ell}}>d$.进而, 如果$X$的局部时$L(x, I)$存在, 那么$L(x, I)$有下面的表现形式

$L(x, I)=(2\pi)^{-d}\int_{\mathbb{R}^d}e^{-i\langle y, x\rangle}\int_{I} e^{i\langle y, X(t)\rangle}\mathrm{d}t\mathrm{d}y.$

(3.3)

特别的, 如果$\sum\limits_{\ell=1}^N\big(\frac{1}{H_{\ell}K_{\ell}}\big)>d$, 那么$B^{H_1, K_1}$和$B^{H_2, K_2}$有相遇局部时存在, 可以写成$L(0, I)$.

证 $X$的占有测度$\mu_I$的傅里叶变换为

$\hat{\mu}=\int_T\exp i\langle\xi, X(t)\rangle\mathrm{d}t.$

根据Fubini定理和引理2.3的性质(ⅰ)可得

$\begin{aligned} \mathcal{J}&=\mathbb{E}\int_{\mathbb{R}^d}|\hat{\mu}(\xi)|^2\mathrm{d}\xi\\ &=\int_{I^2}\int_{\mathbb{R}^d}|\mathbb{E}\exp\big(i\langle y, X(t)-X(s)\rangle\big)|dydtds\\ &=c\int_{I^2}\frac{1}{\big[\mathbb{E}\big(X_0(t)-X_0(s)\big)^2\big]^{d/2}}\\ &\leq\int_{I^2}\frac{dsdt}{(\sum\limits_\limits{\ell=1}^N|t_{\ell}-s_{\ell}|^{2H_{\ell}K_{\ell}})^{d/2}}. \end{aligned}$

(3.4)

根据Plancherel定理, $X$有局部时$L(\cdot, I)\in L^2(\mathbb{P}\times\lambda_d)$, 当且仅当$\mathcal{J}<\infty$.进而由引理2.4可知, $\mathcal{J}<\infty$的充要条件是$\sum\limits_{\ell=1}^N\frac{1}{H_{\ell}K_{\ell}} >d$.类似于定理3.1^[3], 取$x=0$时, 则$B^{H_1, K_1}$和$B^{H_2, K_2}$的相遇局部时$L(0, I)$存在.

定理3.2 设$B^{H_1, K_1}=\{B^{H_1, K_1}(t), t\in\mathbb{R}^N\}$和$B^{H_2, K_2}=\{B^{H_2, K_2}(t), t\in\mathbb{R}^N\}$是两个相互独立的$(N, d)$双分数布朗运动, 其指标分别为$H_1, K_1$; $H_2, K_2$, 则当

$\sum\limits_{\ell=1}^{\tau-1}\frac{1}{H_\ell K_\ell}\leq d<\sum\limits_\limits{\ell=1}^{\tau}\frac{1}{H_\ell K_\ell}$

时, 其中$\tau=\{1, \cdots, N\}$, $B^{H_1, K_1}$和$B^{H_2, K_2}$在$\mathbb{R}^d$上的相遇局部时几乎必然连续.

为了证明定理3.2, 我们首先给出以下两个引理.

引理3.3 假设定理3.1条件成立, 则存在有限正常数$\varepsilon\in(0, 1)$和只依赖于$H_1, K_1, H_2, K_2, N, d$的有限正常数$c_{3.1}$, 使得对所有的$r\in(0, \varepsilon)$, $T=[0, r]^N\subseteq I$, $x\in\mathbb{R}^d$和整数$n\geq1$, 有

${\mathbb{E}}[L(x, T)^n]\leq c_{3.1}^n(n!)^{N-\beta_\tau}r^{n\beta_\tau}.$

(3.5)

证 根据(3.1) 式且$X_1, \cdots, X_d$是相互独立的, 且与$X_0$同分布, 当整数$n>1$时, 有

$\begin{aligned} {\mathbb{E}}[L(x, T)^n] =&{(2\pi)}^{-nd}\int_{\mathbb{R}^{nd}}\int_{T^n}\exp\bigg({-i\sum\limits_{j=1}^{n}\langle u^j, x\rangle}\bigg) \times\mathbb{E}\exp\bigg({i\sum\limits_{j=1}^{n}\langle u^j, X(t^j)\rangle}\bigg)\mathrm{d}\bar{u}\mathrm{d}\bar{t}\\ \leq&(2\pi)^{-nd}\int_{T^n}\prod\limits_{k=1}^{d}\bigg\{\int_{\mathbb{R}^n}\exp\bigg[-\frac{1}{2}{Var} \bigg(\sum\limits_{j=1}^{n}u_k^jX(t^j)\bigg)\bigg]\mathrm{d}\bar{u}_k\bigg\}\mathrm{d}\bar{t}\\ =&(2\pi)^{-\frac{nd}{2}}\int_{T^n}\big[{\det\textrm{Cov}}(X(t^1), \cdots, X(t^n))\big]^{-\frac{d}{2}}\mathrm{d}\bar{t}. \end{aligned}$

(3.6)

其中$\bar{u}_k=(u_k^1, \cdots, u_k^n)\in\mathbb{R}^n, k=1, \cdots, d$.在(3.6) 式中的等号成立是根据下面公式, 即对任意$n\times n$正定矩阵$\Gamma$, 有

$\int_{\mathbb{R}^n}\frac{[\det(\Gamma)]^{1/2}}{(2\pi)^{n/2}}\exp\bigg(-\frac{1}{2}x'\Gamma x\bigg)\mathrm{d}x=1.$

(3.7)

在引理2.7中令$\delta=n^{-1}, q=d$, 即可得到满足(2.18) 和(2.19) 式的$\tau$个大于1的正数$p_1, \cdots, p_\tau$.由Hölder不等式可得

$\begin{aligned} \mathcal{J}_k=&\big[\det\textrm{Cov}\big(X_0(t^1), \cdots, X_0(t^n)\big)\big]^{-\frac{1}{2}}\\ =&\prod\limits_{\ell=1}^{\tau}\big[\det\textrm{Cov}\big(X_0(t^1), \cdots, X_0(t^n)\big)\big]^{-\frac{1}{2p_\ell}}. \end{aligned}$

(3.8)

对任意的点$t^1\cdots t^n\in T$, 且$t_\ell^1\cdots t_\ell^n (1\leq\ell\leq N)$都互不相同, 由(3.6) 和(3.8) 式可以得到

${\mathbb{E}}[L(x, T)^n] \leq (2\pi)^{-\frac{nd}{2}}\int_{T^n}\prod\limits_{\ell=1}^{\tau}\big[\det \textrm{Cov}\big(X_0(t^1), \cdots, X_0(t^n)\big)\big]^{-\frac{d}{2p_\ell}}\mathrm{d}\bar{t}.$

(3.9)

为了求(3.9) 式积分, 我们首先对下标为$\ell=1, \cdots, \tau$求$[\mathrm{d}t_\ell^1\cdots\mathrm{d}t_\ell^n]$的积分.事实上, 对任意的$\ell\in\{1, \cdots, \tau\}$和$t_1, \cdots, t_n\in T=\prod\limits_{\ell=1}^N[a_\ell, a_\ell+r_\ell]$, 有

$a_\ell\leq t_\ell^{\pi_\ell(1)}\leq t_\ell^{\pi_\ell(2)}\leq, \cdots, \leq t_\ell^{\pi_\ell(n)}\leq a_\ell+r_\ell$

(3.10)

其中$\pi_\ell$是$\{1, \cdots, N\}$的一个置换, 根据引理2.3和(2.13) 可得

$\det\textrm{Cov}\big(X_0(t^1), \cdots, X_0(t^n)\big)\geq c_{3, 2}\prod\limits_{j=1}^n\big(t_\ell^{\pi_\ell(j)}-t_\ell^{\pi_\ell(j-1)}\big)^{2H_{\ell}K_{\ell}}.$

(3.11)

记$t_\ell^{\pi_\ell(0)}:=\varepsilon$, 且$\varepsilon<\frac{1}{2}\min\{a_\ell, 1\leq\ell\leq N\}$, 由(3.10) 和(3.11) 式得

$\begin{aligned} &\int_{[a_{\ell}, a_{\ell}+r_{\ell}]^n}\big[\det \textrm{Cov}\big(X_0(t^1), \cdots, X_0(t^n)\big)\big]^{-\frac{d}{2p_\ell}}\mathrm{d}t_\ell^1\cdots\mathrm{d}t_\ell^n\\ \leq& \sum\limits_\limits{\pi_\ell}c^n\int_{a_\ell\leq t_\ell^{\pi_\ell(1)}\leq, \cdots, \leq t_\ell^{\pi_\ell(n)}\leq a_\ell+r_\ell} \prod\limits_{j=1}^n\frac{1}{\big(t_\ell^{\pi_\ell(j)}-t_\ell^{\pi_\ell(j-1)}\big)^{H_{\ell}K_{\ell}d/p_\ell}}\mathrm{d}t_\ell^1 \cdots\mathrm{d}t_\ell^n\\ \leq& c_{3, 3}^n(n!)^{H_{\ell}K_{\ell}d/p_\ell}r_{\ell}^{n\big(1-(1-n)H_{\ell}K_{\ell}d/p_\ell\big)}. \end{aligned}$

(3.12)

根据(2.20) 和(3.12) 式, 继续对下标为$\ell=\tau+1, \cdots, N$求$[\mathrm{d}t_\ell^1\cdots\mathrm{d}t_\ell^n]$的积分, 得

${\mathbb{E}}[L(x, T)^n]\leq c_{3, 3}^n(n!)^{\sum\limits_{\ell=1}^{\tau}H_{\ell}K_{\ell}d/p_\ell}\prod\limits_{\ell=1}^{\tau} r_{\ell}^{n\big(1-(1-n^{-1})H_{\ell}K_{\ell}d/p_\ell\big)}\cdot\prod\limits_{\ell=\tau+1}^Nr_{\ell}^n.$

(3.13)

考虑特殊情况, 即当$r_1=r_2=\cdots=r_N=r$, 根据(2.20) 和(3.13) 式, 其中的$\delta=n^{-1}, q=d$, 得

$\begin{aligned} {\mathbb{E}}[L(x, T)^n]&\leq c_{3, 4}^n(n!)^{\sum\limits_{\ell=1}^{\tau}H_{\ell}K_{\ell}d/p_\ell}r^{n\big(N-(1-n^{-1})\sum\limits_{\ell=1}^{\tau} H_{\ell}K_{\ell}d/p_\ell\big)}\\ &\leq c_{3, 1}^n({n!})^{N-\beta_\tau}r^{n\beta_\tau}. \end{aligned}$

(3.14)

记$\beta_\tau=\sum\limits_{\ell=1}^{\tau}\frac{H_{\tau}K_{\tau}}{H_{\ell}K_{\ell}}+N-\tau-H_{\tau}K_{\tau}d$.

引理3.4 假设定理3.1条件成立, 存在只依赖于$H_1, K_1, H_2, K_2, N, d$的有限正常数$c_{3, 5}$, 则对任意的$T=[a, a+\langle r\rangle]\subseteq I$, $r\in(0, 1)$, $x, y\in\mathbb{R}^d$且$|x-y|\leq1$, 所有的偶数$n\geq1$和$\gamma\in(0, 1\wedge\frac{\alpha_\tau}{2\tau})$有

$\mathbb{E}\big[\big(L(x, T)-L(y, T)\big)^n\big]\leq c_{3, 5}^n|x-y|^{n\gamma}(n!)^{N-\beta_\tau+(1+H_{\tau}K_{\tau})\gamma}r^{n(\beta_\tau-H_{\tau}K_{\tau}\gamma)}.$

(3.15)

证设$\gamma\in(0, 1)$是一个足够小的正常数, 对所有的$u\in\mathbb{R}$有下列两个不等式成立,

$\begin{aligned} &|e^{iu}-1|\leq 2^{1-\gamma}|u|^{\gamma}, \quad \quad\quad\forall u\in\mathbb{R}, \\ &|u+v|^{\gamma}\leq|u|^{\gamma}+|v|^{\gamma}, \quad \quad\quad \forall u, v\in\mathbb{R}. \end{aligned}$

(3.16)

对所有的$u^1\cdots u^n$和$x, y\in\mathbb{R}^d$有下列的式子

$\prod\limits_{j=1}^{n}|e^{-i\langle u^j, x\rangle}-e^{-i\langle u^j, y\rangle}|\leq2^{(1-\gamma)n}|x-y|^{n\gamma}{\sum}'\prod\limits_{j-1}^{n}|u^{j, k_j}|^{\gamma},$

(3.17)

其中${\sum}'$是包括了所有的序列$(k_1, \cdots, k_n)\in\{1, \cdots, d\}^n$.

对任意的偶数$n\geq2$, 根据(3.2) 和(3.17) 式得

$\begin{aligned} &\mathbb{E}[(L(x, T)-L(y, T))^n]\\ \leq& (2\pi)^{-nd}2^{(1-\gamma)n}|x-y|^{n\gamma}\\&\times{\sum}'\int_{T^n}\int_{\mathbb{R}^{nd}} \prod\limits_{n=1}^{n}|u^{m, k_m}|^{\gamma}\mathbb{E}\exp\bigg(-i\sum\limits_\limits{j=i}^{n}\langle u^j, X(t^j)\rangle\bigg)\mathrm{d}\bar{u}\mathrm{d}\bar{t}\\ \leq& c_{3, 6}^n|x-y|^{n\gamma}{\sum}'\int_{T^n}\mathrm{d}\bar{t}\\&\times\prod\limits_{n=1}^{n}\bigg\{\int_{\mathbb{R}^{nd}}|u^{m, k_m}|^{n\gamma} \exp\big[-\frac{1}{2}{\hbox{Var}}\big(\sum\limits_\limits{j=1}^{n}\langle u^j, X(t^j)\rangle\big)\big]\mathrm{d}\bar{u}\bigg\}^{1/n}, \end{aligned}$

(3.18)

其中最后一个不等式由Hölder不等式得到.

此时固定一向量$\bar{k}=(k_1, \cdots, k_n)\in\{1, \cdots, d\}^n$和$n$个不同的点$t^1, \cdots, t^n\in T$, 并对所有的$\ell(1\leq\ell\leq N)$, $t_\ell^1, \cdots, t_\ell^n$是互不相同的.记

$\mathcal{M}=\prod\limits_{m=1}^{n}\bigg\{\int_{\mathbb{R}^{nd}}|u^{m, k_m}|^{n\gamma} \exp\big[-\frac{1}{2}{\hbox{Var}}\big(\sum\limits_{j-1}^{n}\langle u^j, X(t^j)\rangle\big)\big]\mathrm{d}\bar{u}\bigg\}^{1/n}.$

(3.19)

由于$X_i (1\leq i\leq d)$是相互独立的, 且与$X_0$同分布, 根据引理2.3知, 随机变量$X_{i}(t^j)(1\leq i\leq d, 1\leq j\leq n)$是线性独立的.因此由引理2.6可以得到

$\begin{aligned} &\int_{\mathbb{R}^{nd}}|u^{m, k_m}|^{n\gamma}\exp\bigg[-\frac{1}{2}Var\bigg(\sum\limits_\limits{j=1}^{n}\langle u_j, X(t^j)\rangle\bigg)\bigg]\mathrm{d}\bar{u}\\ =&\frac{(2\pi)^{(nd-1)/2}}{\big[\det {\hbox{Cov}}\big(X_0(t^1), \cdots, X_0(t^n)\big)\big]^{d/2}}\int_{\mathbb{R}}\big(\frac{v}{\sigma_m}\big)^{n\gamma} e^{-\frac{v^2}{2}}\mathrm{d}v\\ \leq&\frac{c_{3.7}^{n}(n!)^\gamma}{\big[\det {\hbox{Cov}}\big(X_0(t^1), \cdots, X_0(t^n)\big)\big]^{d/2}}\frac{1}{\sigma_m^{n\gamma}}. \end{aligned}$

(3.20)

上式中的$\sigma_m^2$关于$X_{k_m}(t^m)$与$X_{i}(t^j)$的条件方差$(i\neq{k_m}$或$i={k_m}, j\neq m)$, 其中最后一个不等式由Stirling公式得到.此时由(3.19) 和(3.20) 式得

$\mathcal{M}\leq\frac{c_{3, 7}^n(n!)^\gamma}{\big[\det {\hbox{cov}}\big(X_0(t^1), \cdots, X_0(t^n)\big)\big]^{d/2}}\prod\limits_{m=1}^n\frac{1}{\sigma_m^{\gamma}}.$

(3.21)

令$\delta=\frac{1}{n}$和$q=d$, 常数$p_\ell(\ell=1, \cdots, \tau)$如引理2.7所述.注意到当$\gamma\in(0, \frac{\alpha_\tau}{2\tau})$, 存在$\ell_0\in\{1, \cdots, \tau\}$, 使得

$\frac{H_{\ell_0}K_{\ell_0}d}{p_{\ell_0}}+2H_{\ell_0}K_{\ell_0}\gamma<1.$

(3.22)

根据(3.8) 和(3.21) 式有

$\mathcal{M}\leq c_{3, 8}^n(n!)^\gamma\prod\limits_{\ell=1}^{\tau}\frac{1}{\big[\det {\hbox{cov}}\big(X_0(t^1), \cdots, X_0(t^n)\big)\big]^{d/(2p_\ell)}}\prod\limits_{m=1}^n\frac{1}{\sigma_m^{\gamma}}.$

(3.23)

根据引理2.3得

$\begin{aligned} {\sigma_m^2}&=\mathbf{Var}\big(X_{k_m}(t^m)|X_{k_m}(t^j), j\neq m\big)\\ &\geq c_{3, 9}^n\sum\limits_\limits{\ell=1}^N\min\big\{|t_\ell^m-t_\ell^j|^{2H_\ell K_\ell}\quad j\neq m\big\}. \end{aligned}$

(3.24)

对任意的$n$个点$t^1, \cdots, t^n\in T$, 令$\pi_1, \cdots, \pi_N$是关于$\{1, \cdots, n\}$的$N$个置换, 使得对任意$\ell(1\leq\ell\leq N)$有

$t_\ell^{\pi_\ell(1)}\leq t_\ell^{\pi_\ell(2)}\leq\cdots\leq t_\ell^{\pi_\ell(n)}.$

(3.25)

由(3.24) 和(3.25) 式得

$\begin{aligned} \prod\limits_{m=1}^{n}\frac{1}{\sigma_m^\gamma}&\leq c_{3.9}^{-n\gamma}\prod\limits_{m=1}^{n}\frac{1}{\sum\limits_{\ell=1}^N \big[\big(t_{\ell}^{\pi_{\ell}(m)}-t_{\ell}^{\pi_{\ell}(m-1)}\big)\wedge\big(t_{\ell}^{\pi_{\ell}(m+1)}-t_{\ell}^{\pi_{\ell}(m)}\big)\big]^{H_{\ell}K_{\ell}\gamma}}\\ &\leq c_{3, 10}^n\prod\limits_{m=1}^{n}\frac{1}{\big[\big(t_{\ell_0}^{\pi_{\ell_0}(m)}-t_{\ell_0}^{\pi_{\ell_0}(m-1)}\big)\wedge \big(t_{\ell_0}^{\pi_{\ell_0}(m+1)}-t_{\ell_0}^{\pi_{\ell_0}(m)}\big)\big]^{H_{\ell}K_{\ell}\gamma}}\\ &\leq c_{3.11}^n\prod\limits_{m=1}^{n}\frac{1}{\big(t_{\ell}^{\pi_{\ell_0}(m)}-t_{\ell_0}^{\pi_{\ell_0}(m-1)}\big)^{q_{\ell_0}^{m}H_{\ell_0}K_{\ell_0}\gamma}}, \end{aligned}$

(3.26)

其中$(q_{\ell_0}^1, \cdots, q_{\ell_0}^n)\in\{0, 1, 2\}^n$, $\sum\limits_{m=1}^{n}q_{\ell_0}^m=n, q_{\ell_0}^1=0$, 这样只需考虑$\sigma_m$在下标为$\ell_0$的值.

由(3.23) 和(3.26) 式可得

$\begin{aligned} \int_{T^n}\mathcal{M}(\bar{k}, \bar{t}, \gamma)\mathrm{d}\bar{t}\leq &c_{3.11}^n(n!)^\gamma\int_{T^n}\prod\limits_{\ell=1}^{\tau}\frac{1}{\big[\det {\hbox{cov}}\big(X_0(t^1), \cdots, X_0(t^n)\big)\big]^{\frac{d}{2p_\ell}}}\\ &\times\prod\limits_{m=1}^{n}\frac{1}{\big(t_{\ell_0}^{\pi_{\ell_0}(m)}-t_{\ell_0}^{\pi_{\ell_0}(m-1)}\big)^{q_{\ell_0}^mH_{\ell_0}K_{\ell_0}\gamma}} \mathrm{d}\bar{t}. \end{aligned}$

(3.27)

为求(3.27) 式, 先对下标$\ell=1, \cdots, \tau$求$[\mathrm{d}t_{\ell}^1\cdots\mathrm{d}t_{\ell}^n]$的积分, 首先考虑$\ell=\ell_0$, 由(2.15), (3.11), (3.22) 式得

$\begin{aligned} &\int_{[a_\ell, a_\ell+r_\ell]^n}\frac{1}{\big[\det {\hbox{cov}}\big(X_0(t^1), \cdots, X_0(t^n)\big)\big]^{\frac{d}{2p_\ell}}} \times\prod\limits_{m=1}^{n}\frac{1}{\big(t_{\ell_0}^{\pi_{\ell_0}(m)}-t_{\ell_0}^{\pi_{\ell_0}(m-1)}\big)^{q_{\ell_0}^mH_{\ell_0}K_{\ell_0}\gamma}}\mathrm{d}\bar{t}\\ \leq&\sum\limits_\limits{\pi_{\ell_0}}c^n\int_{a_{\ell_0}\leq t_\ell^{\pi_{\ell_0}(1)}\leq, \cdots, \leq t_{\ell_0}^{\pi_{\ell_0}(n)}\leq a_{\ell_0}+r_{\ell_0}}\\&\times\prod\limits_{j=1}^n\frac{1}{\big(t_{\ell_0}^{\pi_{\ell_0}(j)}-t_{\ell_0}^{\pi_{\ell_0}(j-1)}\big)^{H_{\ell_0}K_{\ell_0}d/p_{\ell_0} +q_{\ell_0}^mH_{\ell_0}K_{\ell_0}\gamma}}\mathrm{d}t_{\ell_0}^1\cdots\mathrm{d}t_{\ell_0}^n\\ \leq& c_{3, 11}^n(n!)^{H_{\ell_0}K_{\ell_0}d/p_{\ell_0}+H_{\ell_0}K_{\ell_0}\gamma} r_{\ell_0}^{n[1-(1-n^{-1})H_{\ell_0}K_{\ell_0}d/p_{\ell_0}-H_{\ell_0}K_{\ell_0}\gamma]}. \end{aligned}$

(3.28)

与此同时, 当$\ell\neq\ell_0$, 由(3.6) 和(3.14) 式得

$\begin{aligned} &\int_{[a_{\ell}, a_{\ell}+r_{\ell}]^n}\big[\det {\hbox{cov}}\big(X_0(t^1), \cdots, X_0(t^n)\big)\big]^{-\frac{d}{2p_\ell}}\mathrm{d}t_\ell^1\cdots\mathrm{d}t_\ell^n\\ \leq& c_{3, 12}^n(n!)^{H_{\ell}K_{\ell}d/p_\ell}r_{\ell}^{n\big(1-(1-n)H_{\ell}K_{\ell}d/p_\ell\big)}. \end{aligned}$

(3.29)

最后继续对下标$\ell=[\tau+1, \cdots, N]$求$[\mathrm{d}t_{\ell}^1\cdots\mathrm{d}t_{\ell}^n]$的积分,

$\begin{aligned} &\int_{T^n}\mathcal{M}(\bar{k}, \bar{t}, \gamma)\mathrm{d}\bar{t} \leq c_{3, 13}^n(n!)^{\sum\limits_{\ell=1}^{\tau}H_{\ell}K_{\ell}/p_{\ell}+H_{\ell_0}K_{\ell_0}\gamma+\gamma}\\ &\times r_{\ell_0}^{n[1-(1-n^{-1})H_{\ell_0}K_{\ell_0}d/p_{\ell_0}-H_{\ell_0}K_{\ell_0}\gamma]}\times \prod\limits_{\ell\neq\ell_0}^{\tau}r_{\ell}^{n\big(1-(1-n)H_{\ell}K_{\ell}d/p_\ell\big)}\times\prod\limits_{\ell=\tau+1}^Nr_\ell^n. \end{aligned}$

(3.30)

特别的, 当$r_1=\cdots=r_N=r<1$时, 综合(3.18) 和(3.30) 式可得

$\begin{aligned} &\mathbb{E}[(L(x, D)-L(y, D))^n]\\ \leq& c_{3, 13}^n|x-y|^{n\gamma}(n!)^{\sum\limits_{\ell=1}^{\tau}H_{\ell}K_{\ell}/p_{\ell}+H_{\ell_0}K_{\ell_0}\gamma+\gamma}\cdot r^{n[N-(1-n^{-1})\sum\limits_{\ell=1}^{\tau}H_{\ell}K_{\ell}d/p_{\ell}-H_{\ell_0}K_{\ell_0}\gamma]}\\ \leq& c_{3, 14}^n|x-y|^{n\gamma}(n!)^{N-\beta_\tau+(1+H_{\tau}K_{\tau})\gamma}r^{n(\beta_\tau-H_{\tau}K_{\tau}\gamma)}, \end{aligned}$

(3.31)

其中$H_{\ell_0}K_{\ell_0}\leq H_{\tau}K_{\tau}$.

定理的3.2证明 令$I\in[\varepsilon, 1]^N$, 其中$\varepsilon>0$, 根据引理3.4, 对任意的$ T\in I, x\in\mathbb{R}^d$, $X$的局部时$L(x, T)$是几乎必然连续的.即对任意的$x, y\in\mathbb{R}^d$和$s, t\in I$, 和任意偶数$n\geq1$, 有

$\begin{aligned} \mathbb{E}\big[\big(L\big(x, [0, t]\big)-L\big(y, [0, s]\big)\big)^n\big] \leq&2^{n-1}\big\{\mathbb{E}\big[\big(L\big(x, [0, t]\big)-L\big(x, [0, s]\big)\big)^n\big]\\&+\mathbb{E}\big[\big(L\big(x, [0, s]\big)-L\big(y, [0, s]\big)\big)^n\big]\big\}, \end{aligned}$

(3.32)

其中$[L(x, (0, t))-L(x, (0, s))]$可以看做是有限个$L(x, T_j)$的和, $T_j\in \mathcal{A}$是$I$上的闭子区间, 其至少有一边长$\leq|s-t|$, 由引理3.3可得

$\mathbb{E}\big[\big(L\big(x, [0, t]\big)-L\big(x, [0, s]\big)\big)^n\big] <c_{3.1}^n(n!)^{N-\beta_\tau}|t-s|^{n\beta_\tau}.$

由引理3.4可得

$\mathbb{E}\big[\big(L\big(x, [0, s]\big)-L\big(y, [0, s]\big)\big)^n\big]< c_{3, 14}^n|x-y|^{n\gamma}(n!)^{N-\beta_\tau+(1+H_{\tau}K_{\tau})\gamma}|t-s|^{n(\beta_\tau-H_{\tau}K_{\tau}\gamma)}, $

即存在足够小的常数$\gamma\in(0, 1)$有

$\mathbb{E}\big[\big(L\big(x, [0, t]\big)-L\big(y, [0, s]\big)\big)^n\big]\leq c_{3, 15}\big(|x-y|+|t-s|\big)^{n\gamma}.$

(3.33)

根据(3.33) 式和多维情形下的Kolmogorov连续定理^[16]得出局部时$L(x, t)$是联合连续的.