# Introduction to dynamical large deviations of Markov processes

## TL;DR

This paper reviews techniques in large deviation theory for analyzing fluctuations of time-additive quantities in Langevin processes, highlighting differences between equilibrium and nonequilibrium systems with detailed examples.

## Contribution

It provides a comprehensive summary of methods for studying dynamical large deviations in Markov processes, including an illustrative example and exercises.

## Key findings

- Large deviation functions characterize fluctuations in Langevin systems.
- Eigenvalue problems are central to calculating fluctuation probabilities.
- Differences between equilibrium and nonequilibrium fluctuations are explained.

## Abstract

These notes give a summary of techniques used in large deviation theory to study the fluctuations of time-additive quantities, called dynamical observables, defined in the context of Langevin-type equations, which model equilibrium and nonequilibrium processes driven by external forces and noise sources. These fluctuations are described by large deviation functions, obtained by solving a dominant eigenvalue problem similar to the problem of finding the ground state energy of quantum systems. This analogy is used to explain the differences that exist between the fluctuations of equilibrium and nonequilibrium processes. An example involving the Ornstein-Uhlenbeck process is worked out in detail to illustrate these methods. Exercises, at the end of the notes, also complement the theory.