# Uncertainty Quantification for Markov Processes via Variational Principles and Functional Inequalities

## TL;DR

This paper develops explicit bounds for uncertainty quantification in Markov processes by combining variational principles with functional inequalities, applicable to steady-states in both discrete and continuous-time settings.

## Contribution

It introduces a novel approach that integrates variational formulas with functional inequalities to derive scalable uncertainty bounds for Markov processes.

## Key findings

- Bounds are well-behaved in the infinite-time limit.
- Applicable to steady-states of discrete and continuous-time Markov processes.
- Provides explicit uncertainty quantification bounds for time-averaged observables.

## Abstract

Information-theory based variational principles have proven effective at providing scalable uncertainty quantification (i.e. robustness) bounds for quantities of interest in the presence of nonparametric model-form uncertainty. In this work, we combine such variational formulas with functional inequalities (Poincar{\'e}, $\log$-Sobolev, Liapunov functions) to derive explicit uncertainty quantification bounds for time-averaged observables, comparing a Markov process to a second (not necessarily Markov) process. These bounds are well-behaved in the infinite-time limit and apply to steady-states of both discrete and continuous-time Markov processes.