# Spectral Properties of Effective Dynamics from Conditional Expectations

## TL;DR

This paper investigates the spectral properties of reduced effective dynamics derived from high-dimensional stochastic systems, providing bounds and numerical insights into their approximation accuracy, especially for reversible and Langevin systems.

## Contribution

It introduces a new relative error bound for the spectrum of effective generators in reversible systems and analyzes spectral approximation methods for Langevin dynamics.

## Key findings

- Eigenfunction approximation error bounds for reversible systems
- KM-type spectral approximation is robust to time window choice
- Effective dynamics can be meaningfully defined for underdamped Langevin systems

## Abstract

The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that if Kramers--Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting.