History-Conditioned Inference of Memory and Stochastic Forcing in Generalized Langevin Equations under Lévy Noise
Abstract
We consider the identification of memory and stochastic forcing in a one-dimensional generalized Langevin equation with state-dependent Brownian noise and optional additive symmetric α-stable jumps. A finite-dimensional Markov embedding provides Kramers--Moyal conditional moment relations that separate memory drift, Brownian diffusion, and jump contributions. From increment ensembles conditioned on the observed velocity history, truncated statistics estimate the stable parameters, corrected second moments determine the Brownian amplitude, and first moments lead to a Volterra equation for the memory kernel, solved by RKHS regularization. We establish conditional finite-sample bounds for the moment estimators and a conditional L2 error bound for kernel recovery with a data-selected regularization parameter. Numerical experiments examine Brownian and Brownian--Lévy models under different forces, memory kernels, observation lengths, measurement noise, and approximate history conditioning.
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BibTeX
@misc{yang2026historyconditioned,
title = {{History-Conditioned Inference of Memory and Stochastic Forcing in Generalized Langevin Equations under Lévy Noise}},
author = {Yang Yang and Ting Gao and Xiaoli Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06879},
eprint = {2610.06879},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06879},
}
RIS
TY - UNPB
TI - History-Conditioned Inference of Memory and Stochastic Forcing in Generalized Langevin Equations under Lévy Noise
AU - Yang Yang
AU - Ting Gao
AU - Xiaoli Chen
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.06879
UR - https://arxiv.org/abs/2610.06879
AB - We consider the identification of memory and stochastic forcing in a one-dimensional generalized Langevin equation with state-dependent Brownian noise and optional additive symmetric α-stable jumps. A finite-dimensional Markov embedding provides Kramers--Moyal conditional moment relations that separate memory drift, Brownian diffusion, and jump contributions. From increment ensembles conditioned on the observed velocity history, truncated statistics estimate the stable parameters, corrected second moments determine the Brownian amplitude, and first moments lead to a Volterra equation for the memory kernel, solved by RKHS regularization. We establish conditional finite-sample bounds for the moment estimators and a conditional L^2 error bound for kernel recovery with a data-selected regularization parameter. Numerical experiments examine Brownian and Brownian--Lévy models under different forces, memory kernels, observation lengths, measurement noise, and approximate history conditioning.
ER -