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One-dimensional Feynman--Kac regularity under the Engelbert--Schmidt conditions

  • Johannes Ruf
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Abstract

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 C1 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 C1,2 in the interior. Thus, in the time-homogeneous one-dimensional setting, continuity suffices for interior C1,2 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.

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Cite

BibTeXRIS
BibTeX
@misc{ruf2026onedimensional,
  title = {{One-dimensional Feynman--Kac regularity under the Engelbert--Schmidt conditions}},
  author = {Johannes Ruf},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29854},
  eprint = {2608.29854},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29854},
}
RIS
TY  - UNPB
TI  - One-dimensional Feynman--Kac regularity under the Engelbert--Schmidt conditions
AU  - Johannes Ruf
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2608.29854
UR  - https://arxiv.org/abs/2608.29854
AB  - 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.
ER  -