Title: Quadratic variation and local times of the horizontal component of the Peano curve (square filling curve) Show Chinese title

Title: 二次变分和Peano曲线（平方填充曲线）水平分量的局部时间

Abstract: We show that the horizontal component of the Peano curve has quadratic variation equal the limit of quadratic variations along the Lebesgue partitions for grids of the form $3^{-n}p\mathbb{Z}+3^{-n}r$, $n=1,2,\ldots$, where $p$ is a rational number, while $r$ is irrational number, but the value of such quadratic variation depends on $p$. This also yields that the horizontal component of the Peano curve is an example of a deterministic function possessing local time (density of the occupation measure) with respect to the Lebesgue measure, whose local time can be expressed as the limit of normalized numbers of interval crossings by this function but the normalization is not a smooth function of the width of the intervals. These two features distinct the horizontal component of the Peano curve from the trajectories of the Wiener process, which is widely used in financial models. Show Chinese abstract

Abstract: 我们证明了皮亚诺曲线的水平分量的二次变差等于在勒贝格分割的网格形式为$3^{-n}p\mathbb{Z}+3^{-n}r$,$n=1,2,\ldots$的二次变差的极限，其中$p$是一个有理数，而$r$是一个无理数，但这种二次变差的值取决于$p$。 这也表明，皮亚诺曲线的水平分量是一个确定性函数的例子，它相对于勒贝格测度具有局部时间（占用测度的密度），其局部时间可以表示为该函数穿过区间的归一化次数的极限，但归一化不是区间宽度的光滑函数。 这两个特性使皮亚诺曲线的水平分量区别于维纳过程的轨迹，维纳过程在金融模型中被广泛使用。