# Exact Distributions of Currents and Frenesy for Markov Bridges

## TL;DR

This paper derives exact formulas for the joint distributions of additive functionals in Markov bridges, with applications to enzymatic reactions, revealing insights into currents, frenesy, and uncertainty relations.

## Contribution

It provides the first exact finite-time distributions for currents and frenesy in Markov bridges, including cases with irreversible transitions.

## Key findings

- Exact finite-time distributions derived for Markov bridges.
- Frenesy can carry significant information about currents.
- Bridges can violate known uncertainty relations under certain conditions.

## Abstract

We consider discrete-time Markov bridges, chains whose initial and final states coincide. We derive exact finite-time formulae for the joint probability distributions of additive functionals of trajectories. We apply our theory to time-integrated currents and frenesy of enzymatic reactions, which may include absolutely irreversible transitions. We discuss the information that frenesy carries about the currents and show that bridges may violate known uncertainty relations in certain cases. Numerical simulations are in perfect agreement with our theory.