Invariant and Super-median Functions for Perturbed Heat Kernels
Abstract
We study super-median and invariant functions for positive and negative Schrödinger perturbations of non-negative kernels. For positive perturbations, combining Dynkin's formula for the original kernel with Duhamel's formula for the perturbation, we derive perturbed Dynkin-type identities and give integrability conditions ensuring invariance. When the perturbed kernel is given by the perturbation series, we obtain a sufficient condition for a function to be super-median without additional integrability assumptions. We also decompose every finite non-negative solution of Duhamel's equation into the perturbation series and a non-negative solution of the corresponding homogeneous equation. For negative perturbations, we obtain analogous identities under an explicit finiteness condition. We also construct functions satisfying Dynkin's formula using zero-resolvents and compensated kernels. Examples involving Gaussian, fractional, and Bessel kernels illustrate the invariance criteria and the role of non-uniqueness in Duhamel's equation.
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BibTeX
@misc{jakubowski2026invariant,
title = {{Invariant and Super-median Functions for Perturbed Heat Kernels}},
author = {Tomasz Jakubowski and Karol Szczypkowski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24860},
eprint = {2610.24860},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24860},
}
RIS
TY - UNPB
TI - Invariant and Super-median Functions for Perturbed Heat Kernels
AU - Tomasz Jakubowski
AU - Karol Szczypkowski
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24860
UR - https://arxiv.org/abs/2610.24860
AB - We study super-median and invariant functions for positive and negative Schrödinger perturbations of non-negative kernels. For positive perturbations, combining Dynkin's formula for the original kernel with Duhamel's formula for the perturbation, we derive perturbed Dynkin-type identities and give integrability conditions ensuring invariance. When the perturbed kernel is given by the perturbation series, we obtain a sufficient condition for a function to be super-median without additional integrability assumptions. We also decompose every finite non-negative solution of Duhamel's equation into the perturbation series and a non-negative solution of the corresponding homogeneous equation. For negative perturbations, we obtain analogous identities under an explicit finiteness condition. We also construct functions satisfying Dynkin's formula using zero-resolvents and compensated kernels. Examples involving Gaussian, fractional, and Bessel kernels illustrate the invariance criteria and the role of non-uniqueness in Duhamel's equation.
ER -