随机符号Takagi--Landsberg桥的二次变差和$p$阶变差
Quadratic and $p$-th variation of random signed Takagi--Landsberg bridges
- King’s College London（伦敦国王学院）
- University of Waterloo（滑铁卢大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究随机符号Takagi--Landsberg桥的变差性质，证明在H=1/2时二次变差对细分分割具有不变性，并构造反例说明其他指标下临界p阶变差存在多聚点，从而刻画唯一不变指标。
AI中文摘要：
我们研究具有独立Rademacher Faber--Schauder系数的随机符号Takagi--Landsberg桥。在指标$H=1/2$时，我们证明每个固定的确定性细分分割序列，若其网格趋于零，则二次变差几乎必然地且关于$t\in[0,1]$一致地等于$t$。对于不必细分的分割，网格条件$o(1/\log n)$是充分的，且其阶是最优的。我们的证明使用了二次和的算子表示以及Rademacher混沌的矩界。我们还表明，非对称符号可以破坏这种不变性。在每一个$H\ne1/2$的指标处，我们构造一个确定性细分序列，沿该序列，临界$p$次幂和（其中$p=1/H$）几乎必然地具有两个不同的有限聚点。在$H=1/4$时，均匀三分网格提供了对二进极限的显式替代。因此，$H=1/2$是对称族中唯一使得临界变差沿每个固定确定性细分序列不变的指标。尽管如此，变差指标在大量分割序列上几乎必然等于$1/H$。
英文摘要：
We study random signed Takagi--Landsberg bridges with independent Rademacher Faber--Schauder coefficients. At index $H=1/2$, we prove that every fixed deterministic refining sequence of partitions with vanishing mesh yields quadratic variation $t$, almost surely and uniformly in $t\in[0,1]$. For partitions that need not be refining, the mesh condition $o(1/\log n)$ is sufficient, and its order is sharp. Our proofs use an operator representation of the quadratic sums and moment bounds for Rademacher chaos. We also show that asymmetric signs can destroy this invariance. At every index $H\ne1/2$, we construct a deterministic refining sequence along which the critical $p$-power sums, with $p=1/H$, have two distinct finite accumulation points almost surely. At $H=1/4$, uniform thirds grids provide an explicit alternative to the dyadic limit. Thus $H=1/2$ is the unique index in the symmetric family at which critical variation is invariant along every fixed deterministic refining sequence. Nevertheless the variation index equals $1/H$ almost surely across a large sequence of partitions.
