AI 中文总结
该研究针对带迭代对数校正的临界分布漂移随机微分方程，通过光滑逼近构造弱解，结合多种估计证明律的唯一性，还得到确定初值解的时间边缘分布密度的双侧高斯估计。
AI 中文摘要
我们研究定义在$\boldsymbol{\rm R}^d$上的随机微分方程$d X_t=b(t,X_t)d t+\boldsymbol{\rm \tiny 2}d W_t$，其中$b$是具有临界Hölder–Besov正则性$-1$的依赖时间、无散度的分布漂移项，且带有迭代对数校正项。对每个初始概率律，我们通过光滑逼近构造弱解，并将奇异漂移项实现为可加泛函。主要分析要素是带有对数小因子的Schauder估计，结合一致对数Krylov估计及分布检验函数的随机代换公式，该估计使我们能应用Zvonkin变换并在满足对应Krylov界的弱解中证明律的唯一性。对于从确定点出发的解，我们进一步证明其时间边缘分布具有满足双侧Aronson型高斯估计的密度。
英文摘要
We study the stochastic differential equation $$d X_t=b(t,X_t)d t+\sqrt{2}d W_t$$ on $\mathbb R^d$, where $b$ is a time-dependent, divergence-free distributional drift of critical Hölder--Besov regularity $-1$, strengthened by an iterated-logarithmic correction. For every initial probability law, we construct a weak solution by smooth approximation and realize the singular drift as an additive functional. The main analytic ingredient is the Schauder estimate with a logarithmic smallness factor. Combined with uniform logarithmic Krylov estimates and a stochastic substitution formula for distributional test functions, this estimate allows us to apply a Zvonkin transformation and prove uniqueness in law among weak solutions satisfying the corresponding Krylov bounds. For solutions starting from deterministic points, we further show that their time-marginal distributions admit densities satisfying two-sided Aronson-type Gaussian estimates.
