Large time behavior of Lévy processes and their nonlocal Schrödinger semigroups

Abstract

We study the large time asymptotics of the Feynman-Kac semigroups of the symmetric Lévy process on unbounded open sets. Our main result proves the exact exponential asymptotic decay rate for the survival probability given in terms of the bottom of the spectrum of the associated nonlocal Schrödinger operator. We also prove quantitative upper bounds with an explicit polynomial correction. The proof is probabilistic and is done by decomposing the Lévy process into a finite-range jump process collecting the small jumps and an independent compound Poisson process describing the large jumps. Our approach gives a direct link between the spectral properties of nonlocal Schrödinger operators and pointwise decay of survival probabilities, which was previously unknown for jump processes.

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BibTeX RIS

BibTeX

@misc{kwasnicki2026large,
  title = {{Large time behavior of Lévy processes and their nonlocal Schrödinger semigroups}},
  author = {Mateusz Kwaśnicki and Phanuel Mariano and Hugo Panzo and Jing Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05698},
  eprint = {2610.05698},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05698},
}

RIS

TY  - UNPB
TI  - Large time behavior of Lévy processes and their nonlocal Schrödinger semigroups
AU  - Mateusz Kwaśnicki
AU  - Phanuel Mariano
AU  - Hugo Panzo
AU  - Jing Wang
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.05698
UR  - https://arxiv.org/abs/2610.05698
AB  - We study the large time asymptotics of the Feynman-Kac semigroups of the symmetric Lévy process on unbounded open sets. Our main result proves the exact exponential asymptotic decay rate for the survival probability given in terms of the bottom of the spectrum of the associated nonlocal Schrödinger operator. We also prove quantitative upper bounds with an explicit polynomial correction. The proof is probabilistic and is done by decomposing the Lévy process into a finite-range jump process collecting the small jumps and an independent compound Poisson process describing the large jumps. Our approach gives a direct link between the spectral properties of nonlocal Schrödinger operators and pointwise decay of survival probabilities, which was previously unknown for jump processes.
ER  -

Abstract

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