PreprintCC BY 4.0arXiv
Sharp variational, jump and oscillation bounds in a general Gaussian context
Abstract
We consider a general, nonsymmetric Ornstein--Uhlenbeck semigroup (ℋt)t>0. We prove an Lp bound for the jump quasi-seminorms for 1 < p < ∞ and a weak type (1,1) oscillation inequality, both with respect to the invariant measure. These results are established for the order ϱ=2. To do so, we analyze specific components of (ℋt)t>0, by distinguishing between small and large values of t, and between local and global spatial zones. This decomposition allows us to explicitly identify which parts of the semigroup remain bounded and which are responsible for the failure of boundedness, both in a weak and in a strong sense, even with respect to Lebesgue measure.
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BibTeX
@misc{casarino2026sharp,
title = {{Sharp variational, jump and oscillation bounds in a general Gaussian context}},
author = {Valentina Casarino and Paolo Ciatti and Peter Sjögren},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19959},
eprint = {2609.19959},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19959},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp variational, jump and oscillation bounds in a general Gaussian context
AU - Valentina Casarino
AU - Paolo Ciatti
AU - Peter Sjögren
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19959
UR - https://arxiv.org/abs/2609.19959
AB - We consider a general, nonsymmetric Ornstein--Uhlenbeck semigroup (ℋ_t)_t>0. We prove an L^p bound for the jump quasi-seminorms for 1 < p < ∞ and a weak type (1,1) oscillation inequality, both with respect to the invariant measure. These results are established for the order ϱ=2. To do so, we analyze specific components of (ℋ_t)_t>0, by distinguishing between small and large values of t, and between local and global spatial zones. This decomposition allows us to explicitly identify which parts of the semigroup remain bounded and which are responsible for the failure of boundedness, both in a weak and in a strong sense, even with respect to Lebesgue measure.
ER -