# A Liouville theorem for stationary and ergodic ensembles of parabolic systems

## TL;DR

This paper proves a Liouville theorem for stationary ergodic parabolic systems and establishes large-scale regularity estimates for caloric functions, advancing understanding of their long-term behavior.

## Contribution

It introduces a first-order Liouville theorem under minimal assumptions and provides almost sure large-scale regularity estimates for caloric functions.

## Key findings

- Liouville theorem holds for stationary ergodic parabolic systems
- Almost sure large-scale $C^{1,eta}$ regularity for caloric functions
- Results apply under qualitative stationarity and ergodicity assumptions

## Abstract

A first-order Liouville theorem is obtained for random ensembles of uniformly parabolic systems under the mere qualitative assumptions of stationarity and ergodicity. Furthermore, the paper establishes, almost surely, an intrinsic large-scale $C^{1,\alpha}$-regularity estimate for caloric functions.