本章导读 本章研究布朗运动的零点集 \(\mathcal Z=\{t:B_t=0\}\) 及其局部时。§7.1 定义（Def 7.1 零点集 \(\mathcal Z\) 与 \(\mathcal Z_t\equiv[0,t]\cap\mathcal Z\)）。§7.2 性质（命题 7.1：以概率 1，\(\mathcal Z\) 无界、闭、且 \(\forall\varepsilon>0\)，\(\mathcal Z\cap(0,\varepsilon)\neq\emptyset\)——证明分别用到 Ex 1.1 的 arctan 命中公式、\(\mathcal Z=B^{-1}(0)\) 的闭性、以及时间反演 \(Y_t=tB_{1/t}\) 仍是 BM）。§7.3 局部时（\(L_t\) 度量到 \(t\) 为止待在零点集的"时间量"；Rmk 7.1：Lebesgue 测度恒为 \(0\) 不可用；用 Hausdorff 测度族——Def 7.2 Minkowski 容度、Def 7.3 用 \(\tfrac12\)-容度定义 \(L_t\)、Def 7.4 等价定义、Rmk 7.2：两定义一致，关键是把零点集测度放大 \(1/\sqrt\varepsilon\)）。无图。

定义 7.1（零点集）/ Definition 7.1 (Zero set) 设 \(\{B_t\}\) 是标准布朗运动。其零点集 \(\mathcal Z\) 为 \(\mathcal Z=\{t:B_t=0\}\)，并定义到 \(t\) 为止的零点集Let \(\{B_t\}\) be a standard Brownian motion. Its zero set \(\mathcal Z\) is \(\mathcal Z=\{t:B_t=0\}\), and define the zero set up to \(t\) by

$$\mathcal Z_t\equiv[0,t]\cap\mathcal Z=\{s:s\le t\text{ and }B_s=0\}.$$

命题 7.1 / Proposition 7.1 以概率 1，(1) \(\mathcal Z\) 无界；(2) \(\mathcal Z\) 是闭集；(3) 对 \(\forall\varepsilon>0\)，\(\mathcal Z\cap(0,\varepsilon)\neq\emptyset\)。With probability one, (1) \(\mathcal Z\) is unbounded; (2) \(\mathcal Z\) is closed; (3) for all \(\varepsilon>0\), \(\mathcal Z\cap(0,\varepsilon)\neq\emptyset\).

命题 7.1 证明 / Proof of Proposition 7.1 (1)：由例 1.1，对任意 \(0(1): by Example 1.1, for any \(0

$$\mathbb P\{B_r=0\text{ for some }s\le r\le a\}=1-\frac2\pi\arctan\!\left(\frac1{\sqrt t}\right),\qquad 1+t=\frac as.$$

故对任意 \(s>0\)，总能找到 \(a>s\) 使得当 \(a\to\infty\)（即 \(t\to\infty\)）时 \(\mathbb P\{B_r=0\text{ for some }s\le r\le a\}\to1\)，这正说明 \(\mathcal Z\) 向上无界。(2)：注意 \(\mathcal Z=B^{-1}(0)\)，其中 \(B:t\mapsto B_t\) 连续，故 \(\mathcal Z\) 是闭集。(3)：对任意小 \(\varepsilon>0\)，\(Y_t=tB_{1/t}\) 仍是标准布朗运动（脚注 7.1：逐条验证标准 BM 条件即得），故对 \(t\ge t_\varepsilon\)（\(t_\varepsilon=\tfrac2\varepsilon\)）有 \(\mathbb P\{\mathcal Z\cap(0,\varepsilon)\neq\emptyset\}=\mathbb P\{Y_t=0\text{ for some }t\ge t_\varepsilon\}\)，由 (1) 此概率等于 1。\(\blacksquare\)So for any \(s>0\) we can always find \(a>s\) such that \(\mathbb P\{B_r=0\text{ for some }s\le r\le a\}\to1\) as \(a\to\infty\) (i.e. \(t\to\infty\)), which exactly shows \(\mathcal Z\) is unbounded above. (2): note \(\mathcal Z=B^{-1}(0)\) where \(B:t\mapsto B_t\) is continuous, so \(\mathcal Z\) is closed. (3): for any arbitrarily small \(\varepsilon>0\), \(Y_t=tB_{1/t}\) is a standard Brownian motion (footnote 7.1: obvious by checking the conditions for standard BM), so for \(t\ge t_\varepsilon\) (with \(t_\varepsilon=\tfrac2\varepsilon\)), \(\mathbb P\{\mathcal Z\cap(0,\varepsilon)\neq\emptyset\}=\mathbb P\{Y_t=0\text{ for some }t\ge t_\varepsilon\}\), which equals 1 by (1). \(\blacksquare\)

局部时的动机 / Motivation 设 \(\{B_t\}\) 是标准布朗运动，零点集 \(\mathcal Z=\{t:B_t=0\}\)。局部时 \(L_t\) 是对到 \(t\) 为止待在零点集上的"时间量"的某种可行度量。Let \(\{B_t\}\) be a standard Brownian motion with zero set \(\mathcal Z=\{t:B_t=0\}\). The local time \(L_t\) is some viable measure of the amount of time spent in the zero set up to \(t\).

注 7.1（为何不用 Lebesgue 测度）/ Remark 7.1 (Why not the Lebesgue measure) \(L_t\) 不能取 Lebesgue 测度（脚注：Lebesgue 测度是 \(n\) 维欧氏空间中度量子集的标准方式，\(n=1,2,3\) 时即长度、面积、体积），因为在 Lebesgue 测度下它的期望恒为 \(0\)。记 \(L_t\) 的 Lebesgue 测度为 \(\lambda(L_t)\)，则\(L_t\) cannot be the Lebesgue measure (footnote: the Lebesgue measure is the standard way of measuring subsets in \(n\)-dimensional Euclidean space; for \(n=1,2,3\) it is length, area, volume), since under it the expected value is always \(0\). Denoting the Lebesgue measure of \(L_t\) by \(\lambda(L_t)\), then

$$\mathbb E[\lambda(L_t)]=\mathbb E\!\left[\int_0^1\mathbf 1\{B_s=0\}\,ds\right]=\int_0^1\mathbb E[\mathbf 1\{B_s=0\}]\,ds=\int_0^1\underbrace{\mathbb P\{B_s=0\}}_{=\,0}\,ds=0,$$

其中用到交换积分次序。因此局部时改用 Hausdorff 测度族来定义。where we switch the order of integration. Hence local time is instead defined within the family of Hausdorff measures.

局部时的理想性质 / Desired properties of local time 局部时 \(L_t\) 应满足：(i) \(L_t\) 是 \(t\) 的随机函数（脚注 7.2：随机性来自 \(\{B_s:s\le t\}\)）；(ii) \(L_t\) 关于 \(t\) 连续；(iii) \(L_t\) 关于 \(t\) 弱递增——若 \(sLocal time \(L_t\) should satisfy: (i) \(L_t\) is a random function of \(t\) (footnote 7.2: the randomness comes from \(\{B_s:s\le t\}\)); (ii) \(L_t\) is continuous in \(t\); (iii) \(L_t\) is weakly increasing in \(t\) — if \(s

定义 7.2、7.3（Minkowski 容度与局部时）/ Definitions 7.2, 7.3 为得到上述性质，令 \(\mathrm{distance}(\cdot)\) 与 \(\mathrm{volume}(\cdot)\) 分别为 \(n\) 维距离与 \(n\) 维体积的 Lebesgue 测度。定义 7.2（Minkowski 容度）：设 \(V\) 是 \(\mathbb R^d\) 的紧子集，则集合 \(V\) 的 \(\alpha\) 维 Minkowski 容度定义为To obtain the above properties, let \(\mathrm{distance}(\cdot)\) and \(\mathrm{volume}(\cdot)\) be the Lebesgue measure of \(n\)-dimensional distance and \(n\)-dimensional volume. Definition 7.2 (Minkowski content): let \(V\) be a compact subset of \(\mathbb R^d\); then the \(\alpha\)-dimensional Minkowski content of \(V\) is defined by

$$\lim_{\varepsilon\downarrow0}\varepsilon^{d-\alpha}\,\mathrm{volume}\{Z:\mathrm{distance}(Z,V)\le\varepsilon\}.$$

定义 7.3（局部时）：把 \(V\) 的 \(\alpha\) 维 Minkowski 容度记为 \((\alpha\text{-cont})\)。设 \(\{B_t\}\) 是标准布朗运动，\(\mathcal Z_t\equiv[0,t]\cap\mathcal Z\)。则局部时 \(L_t\) 形式上定义为Definition 7.3 (Local time): denote the \(\alpha\)-dimensional Minkowski content of \(V\) by \((\alpha\text{-cont})\). Let \(\{B_t\}\) be a standard Brownian motion and \(\mathcal Z_t\equiv[0,t]\cap\mathcal Z\). Then the local time \(L_t\) is formally defined by

$$L_t\equiv\sqrt{\frac\pi4}\left(\tfrac12\text{-cont}\right)(\mathcal Z_t)=\sqrt{\frac\pi4}\,\lim_{\varepsilon\downarrow0}\varepsilon^{1-\frac12}\int_0^t\mathbf 1\{\mathcal Z_t\cap[s-\varepsilon,s+\varepsilon]\neq\emptyset\}\,ds.$$

定义 7.4 与注 7.2（等价定义）/ Definition 7.4 and Remark 7.2 定义 7.4（局部时的等价定义）：设 \(\{B_t\}\) 是标准布朗运动。令 \(\mathbf 1\{x,\varepsilon\}\) 为指示函数，当存在 \(t\in[x-\varepsilon,x+\varepsilon]\) 使 \(B_t=0\) 时取值 \(1\)。令 \(L_{t,\varepsilon}=\dfrac{\sqrt\pi}{2\sqrt\varepsilon}\displaystyle\int_0^t\mathbf 1\{s,\varepsilon\}\,ds\)，再定义局部时 \(L_t=\lim_{\varepsilon\downarrow0}L_{t,\varepsilon}\)。注 7.2：两个定义完全相同，都要求把零点集的测度放大 \(\dfrac1{\sqrt\varepsilon}\) 倍，使零点集的新测度不再为零。Definition 7.4 (Alternative definition of local time): let \(\{B_t\}\) be a standard Brownian motion. Let \(\mathbf 1\{x,\varepsilon\}\) be the indicator function that takes value \(1\) when \(B_t=0\) for some \(t\in[x-\varepsilon,x+\varepsilon]\). Let \(L_{t,\varepsilon}=\dfrac{\sqrt\pi}{2\sqrt\varepsilon}\displaystyle\int_0^t\mathbf 1\{s,\varepsilon\}\,ds\), and then define the local time \(L_t=\lim_{\varepsilon\downarrow0}L_{t,\varepsilon}\). Remark 7.2: the two definitions are exactly the same, both requiring that the measure of the zero set be enlarged by \(\dfrac1{\sqrt\varepsilon}\) so that the new measure of the zero set is non-zero anymore.