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arXiv · 1807.06543

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

Bernard Derrida·Tridib Sadhu
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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 $Q_T$.

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BibTeXRIS

Bernard Derrida, Tridib Sadhu. 2018-07-17. Large deviations conditioned on large deviations I: Markov chain and Langevin equation. https://doi.org/10.1007/s10955-019-02321-4

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