Engelbert–Schmidt条件下的一维Feynman–Kac正则性

中文摘要 AI 辅助

设$X$为系数满足Engelbert–Schmidt条件的时齐一维扩散过程，$q$为有界位势。我们证明，相关的Feynman–Kac半群在时间上光滑、在空间上为$C^1$，其一阶空间导数局部绝对连续，Kolmogorov方程几乎处处成立。若漂移项、二阶系数与位势连续，则Feynman–Kac值在内部为$C^{1,2}$，说明在时齐一维情形下，连续性足以保证内部$C^{1,2}$正则性。我们将对应半群与被杀死、反射的Feynman–Kac泛函关联，处理非零、与时间无关的Dirichlet数据，并给出实例说明连续性假设的尖锐性及该结论在二维情形下不成立。

英文摘要

Let $X$ be a time-homogeneous one-dimensional diffusion whose coefficients satisfy the Engelbert--Schmidt conditions, and let $q$ be a bounded potential. We show that the associated Feynman--Kac semigroup is smooth in time and $C^1$ in space, with a locally absolutely continuous first spatial derivative; the Kolmogorov equation holds almost everywhere. If the drift, second-order coefficient, and potential are continuous, the Feynman--Kac value is $C^{1,2}$ in the interior. Thus, in the time-homogeneous one-dimensional setting, continuity suffices for interior $C^{1,2}$ regularity. We identify the corresponding semigroups with the killed and reflected Feynman--Kac functionals, treat nonzero, time-independent Dirichlet data, and give examples showing the sharpness of the continuity assumptions and the failure of the corresponding statement in two dimensions.

发表机构

• London School of Economics and Political Science（伦敦政治经济学院）

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