Large deviations conditioned on large deviations I: Markov chain and Langevin equation

TL;DR

This paper systematically analyzes how stochastic processes like Markov chains and Langevin equations behave when conditioned on large deviations of their empirical measures over long time intervals, deriving modified dynamics and fluctuation properties.

Contribution

It introduces a unified framework for analyzing conditioned large deviations in Markov processes and Langevin dynamics, including new methods for calculating conditioned large deviation functions.

Findings

Derived conditioned large deviation functions for Langevin dynamics.

Calculated typical trajectories and fluctuations under conditioning.

Extended results from discrete Markov chains to continuous processes.

Abstract

We present a systematic analysis of stochastic processes conditioned on an empirical measure $Q_T$ defined in a time interval $[0,T]$ for large $T$. We build our analysis starting from a discrete time Markov chain. Results for a continuous time Markov process and Langevin dynamics are derived as limiting cases. We show how conditioning on a value of $Q_T$ modifies the dynamics. For a Langevin dynamics with weak noise, we introduce conditioned large deviations functions and calculate them using either a WKB method or a variational formulation. This allows us, in particular, to calculate the typical trajectory and the fluctuations around this optimal trajectory when conditioned on a certain value of $Q_T$.