Error analysis beyond exponential kernels for Prony-series approximations

Develop rigorous error analysis for generalized Langevin equations when the memory kernel is approximated by a Prony series, beyond the simple one-dimensional harmonic oscillator case with exponentially decaying kernels; in particular, derive trajectory-level error bounds for more general kernels and settings.

Background

In prior work on kernel identification and approximation, Prony-series representations have been analyzed for a one-dimensional harmonic oscillator under exponentially decaying kernels, yielding error bounds in that restricted setting.

The present paper highlights that assuming exponential decay is a strong limitation and explicitly states that extending error analysis to more general cases remains open, motivating broader theoretical frameworks for non-exponential (e.g., subexponential) kernels and higher-dimensional dynamics.

References

However, the kernel is assumed to have exponential decay, and the error analysis for more general cases remains open.

— Error Analysis of Generalized Langevin Equations with Approximated Memory Kernels  (2512.10256 - Lang et al., 11 Dec 2025) in Introduction — Related works (Error analysis paragraph)

Another open question is the optimal approximation of arbitrary memory kernels with the smallest number of Maxwell bodies $C$.

— Quantum Simulation of Dissipative Non-Markovian Coupled Classical Oscillators  (2609.20721 - Schade et al., 17 Sep 2026) in Section Discussion; Section 7.2, “Kernel Approximation”