发表机构
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, CAS（中国科学院数学与系统科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究临界无散度漂移扩散的Varadhan渐近公式，证明扇形条件下定量下界，给出反例与结构分解上界，并构造满足EVF但路径律与布朗律相互奇异的漂移。
AI 中文摘要
我们研究了在维度$d\geq2$中，具有临界无散度漂移的扩散的欧几里得Varadhan公式（EVF）和路径律。对于由光滑有界反对称流矩阵给出的漂移，我们在一个带有低阶项的扇形条件下证明了定量下界。其常数仅依赖于维度和扇形条件数据。通过逼近，局部一致的欧几里得下界对所有无散度$L^\infty_t{\rm BMO}^{-1}_x$漂移成立。然而，一个依赖于时间的反例表明，即使对于$L^\infty_tL^d_x$中的漂移，EVF也可能失效。在结构分解条件下给出了一个互补的上界。特别地，对于每个与时间无关的$L^{d,\infty}$漂移（无需任何小性假设），以及与二维涡度方程相关的解耦扩散，EVF成立。最后，我们构造了一个与时间无关的$L^d$漂移，其SDE从原点具有唯一强解，且其路径律与相应的无漂移布朗律相互奇异。同一扩散满足EVF，表明尽管路径律具有奇异性，布朗短时端点代价仍可持续存在。
英文摘要
We study the Euclidean Varadhan formula (EVF) and path laws for diffusions with critical divergence-free drifts in dimension $d\geq2$. For drifts given by smooth bounded skew-symmetric stream matrices, we prove a quantitative lower bound under a sector condition with a lower-order term. Its constants depend only on the dimension and the sector condition data. By approximation, the locally uniform Euclidean lower bound holds for all divergence-free $L^\infty_t{\rm BMO}^{-1}_x$ drifts. However, a time-dependent counterexample shows that EVF can fail even for a drift in $L^\infty_tL^d_x$. A complementary upper bound is given under a structural decomposition condition. In particular, for every time-independent $L^{d,\infty}$ drift (without any smallness assumption), and the decoupled diffusion associated with the two-dimensional vorticity equation, the EVF holds. Finally, we construct a time-independent $L^d$ drift whose SDE has a unique strong solution from the origin and whose path law is mutually singular with the corresponding driftless Brownian law. The same diffusion satisfies EVF, showing that Brownian short-time endpoint costs can persist despite singularity of the path law.
Comments37 pages, 2 figures
