AI 中文总结
该研究在有界解析若尔当域中证明了击中弦形$\text{SLE}_2$的小布朗环的根强度测度经$ε^{5/4}$缩放后依$L^1$收敛到自然含量测度的常数倍，给出精确极限常数，并得到淡收敛结果与布朗环汤的大数定律应用。
AI 中文摘要
设$D$为有界解析若尔当域，$γ$为$D$中的弦形$\text{SLE}_2$，赋予其$5/4$维自然含量测度$μ_γ$。我们保留布朗环测度的标准积分布朗桥表示中的根，令$\text{mathcal M}_ε^γ$为持续时间在$[ε^2,t_0]$内、迹击中$γ$的环的根强度测度。对每个$f∈C_c(D)$，我们证明$ε^{5/4}\text{mathcal M}_ε^γ(f)$在$L^1$中收敛于$\frac{4}{5π}\bar v_{\text{BB}}μ_γ(f)$，其中$\bar v_{\text{BB}}∈(0,∞)$是自然时间双侧全平面$\text{SLE}_2$的布朗桥偏移扫过的平均比面积。特别地，正随机测度依概率淡收敛。证明用到了持续时间倍频程恒等式、由停时双臂马尔可夫骨架得到的确定性有限$R$参考系数、带标记的物理Palm切、环形远返回估计以及合法介观对角线。作为应用，独立布朗环汤的均匀时间标记根满足相应的淡大数定律。解析边界假设仅通过全局有限域一致可积估计引入；其对任意有界若尔当域的类似结论仍未解决。
英文摘要
Let $D$ be a bounded analytic Jordan domain and let $γ$ be chordal $\mathrm{SLE}_2$ in $D$, equipped with its $5/4$-dimensional natural-content measure $μ_γ$. We retain the root in the standard integrated Brownian-bridge representation of Brownian loop measure and let $\mathcal M_\varepsilon^γ$ be the root-intensity measure of loops with duration in $[\varepsilon^2,t_0]$ whose traces hit $γ$. For every $f\in C_c(D)$, we prove that $\varepsilon^{5/4}\mathcal M_\varepsilon^γ(f)$ converges in $L^1$ to $\frac{4}{5π}\bar v_{\mathrm{BB}}μ_γ(f)$, where $\bar v_{\mathrm{BB}}\in(0,\infty)$ is the mean specific area swept out by the Brownian-bridge offset of a natural-time two-sided whole-plane $\mathrm{SLE}_2$. In particular, the positive random measures converge vaguely in probability. The proof uses a duration-octave identity, a deterministic finite-$R$ reference coefficient obtained from a stopped two-arm Markov skeleton, a marked physical Palm tangent, an annular remote-return estimate, and a legal mesoscopic diagonal. As an application, the uniformly time-marked roots of an independent Brownian loop soup satisfy the corresponding vague law of large numbers. The analytic-boundary hypothesis enters only through a global finite-domain uniform-integrability estimate; its analogue for an arbitrary bounded Jordan domain remains open.
CommentsThis paper contains critical errors in key derivations. The authors have decided to withdraw it for substantial revision and resubmission
