Infinite ergodic theory and functional statistics of non-confined Feller process
Abstract
We study additive observables of a non-confined Feller process characterized by a non- normalizable stationary probability density and return times with infinite mean. The process is equivalent, after a deterministic rescaling, to a squared Bessel process, but we focus here on a question: the spatial structure of its local-time field. We show that the local time admits a fac- torization in the long time limit. Its spatial profile is governed by the non-normalizable stationary density, whereas its temporal fluctuations are controlled by a single Mittag-Leffler random ampli- tude. As a consequence, normalized spatial correlations of the local time converge to a universal constant independent of the two observation levels. Occupation times of finite intervals follow as corollaries and display Darling-Kac fluctuations. In contrast, non-integrable power observables exhibit self-similar squared-Bessel-type limits rather than Mittag-Leffler statistics. This spatial- field perspective provides a unifying framework for infinite ergodic theory under state-dependent multiplicative noise.
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BibTeX
@misc{mendez2026infinite,
title = {{Infinite ergodic theory and functional statistics of non-confined Feller process}},
author = {Vicenç Méndez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10106},
eprint = {2609.10106},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10106},
note = {CC0},
}
RIS
TY - UNPB
TI - Infinite ergodic theory and functional statistics of non-confined Feller process
AU - Vicenç Méndez
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10106
UR - https://arxiv.org/abs/2609.10106
AB - We study additive observables of a non-confined Feller process characterized by a non- normalizable stationary probability density and return times with infinite mean. The process is equivalent, after a deterministic rescaling, to a squared Bessel process, but we focus here on a question: the spatial structure of its local-time field. We show that the local time admits a fac- torization in the long time limit. Its spatial profile is governed by the non-normalizable stationary density, whereas its temporal fluctuations are controlled by a single Mittag-Leffler random ampli- tude. As a consequence, normalized spatial correlations of the local time converge to a universal constant independent of the two observation levels. Occupation times of finite intervals follow as corollaries and display Darling-Kac fluctuations. In contrast, non-integrable power observables exhibit self-similar squared-Bessel-type limits rather than Mittag-Leffler statistics. This spatial- field perspective provides a unifying framework for infinite ergodic theory under state-dependent multiplicative noise.
ER -