带奇异漂移的跳跃型随机输运方程的正则性保持
Regularity Preservation for Jump-Type Stochastic Transport Equations with Singular Drift
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中文总结 AI 辅助
该研究针对带奇异漂移的跳跃型随机输运方程，构建随机特征系统并结合多种分析方法，证明了弱可微解的存在唯一性及初始数据一阶Sobolev正则性的保持。
中文摘要 AI 辅助
我们研究由布朗输运噪声和非线性状态依赖泊松跳跃项驱动的一阶随机输运方程。漂移向量场仅满足可积性，且符合次临界Krylov--Röckner条件。跳跃系数对解的依赖在解值与其空间梯度间构建了非平凡耦合。为处理该耦合，我们针对位置、解值和梯度构建了随机特征系统，并借助带跳跃的Itô--Wentzell公式推导特征表示。对于奇异漂移，我们结合光滑逼近、Zvonkin变换、随机流估计及随机Gronwall不等式，以得到弱可微极限。通过对两个解的差进行重正化能量估计（含泊松补偿项的贡献），直接在弱可微类中建立唯一性。在跳跃系数满足适当可积性和可微性假设下，我们证明了弱可微解的存在性与唯一性，还表明初始数据的一阶空间Sobolev正则性在演化过程中得以保持。
英文摘要
We study a first-order stochastic transport equation driven by Brownian transport noise and a nonlinear state-dependent Poisson jump term. The drift vector field is merely integrable and satisfies the subcritical Krylov--Röckner condition. The dependence of the jump coefficient on the solution creates a nontrivial coupling between the solution value and its spatial gradient. To handle this coupling, we construct a stochastic characteristic system for the position, the solution value, and the gradient, and derive a characteristic representation by means of an Itô--Wentzell formula with jumps. For singular drifts, we combine smooth approximation, the Zvonkin transformation, estimates for stochastic flows, and stochastic Gronwall inequalities to pass to a weakly differentiable limit. Uniqueness is established directly in the weakly differentiable class through a renormalized energy estimate for the difference of two solutions, including the contribution of the Poisson compensator. Under suitable integrability and differentiability assumptions on the jump coefficient, we prove the existence and uniqueness of weakly differentiable solutions. We further show that the first-order spatial Sobolev regularity of the initial datum is preserved by the evolution.
