# Stochastic Lagrangians for Statistical Dynamics

## TL;DR

This paper introduces the stochastic Lagrangian formalism as a theoretical framework for analyzing statistical dynamics of systems governed by stochastic differential equations, emphasizing its invariance properties and applications.

## Contribution

It presents a comprehensive review of stochastic Lagrangian formalism and demonstrates its application to physically relevant stochastic dynamical systems.

## Key findings

- Stochastic Lagrangian formalism effectively formulates realization probabilities.
- The approach highlights invariance properties of statistical dynamics.
- Applications to physical systems illustrate the formalism's utility.

## Abstract

The concept of stochastic Lagrangian and its use in statistical dynamics is illustrated theoretically, and with some examples. Dynamical variables undergoing stochastic differential equations are stochastic processes themselves, and their realization probability functional within a given time interval arises from the interplay between the deterministic parts of dynamics and noise statistics. The stochastic Lagrangian is a tool to formulate realization probabilities via functional integrals, once the statistics of noises involved in the stochastic dynamical equations is known. In principle, it allows to highlight the invariance properties of the statistical dynamics of the system. In this work, after a review of the stochastic Lagrangian formalism, some applications of it to physically relevant cases are illustrated.