Journal of Statistical Physics

, Volume 157, Issue 1, pp 182–204 | Cite as

Hamiltonian and Lagrangian for the Trajectory of the Empirical Distribution and the Empirical Measure of Markov Processes

  • Frank Redig
  • Feijia Wang


We compute the Hamiltonian and Lagrangian associated to the large deviations of the trajectory of the empirical distribution for independent Markov processes, and of the empirical measure for translation invariant interacting Markov processes. We treat both the case of jump processes (continuous-time Markov chains and interacting particle systems) as well as diffusion processes. For diffusion processes, the Lagrangian is a quadratic form of the deviation of the trajectory from the solution of the Kolmogorov forward equation. In all cases, the Lagrangian can be interpreted as a relative entropy or relative entropy density per unit time.


Hamiltonian Lagrangian Large deviations Empirical distribution Empirical measure Feng–Kurtz formalism 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Delft Institute of Applied MathematicsTechnische Universiteit DelftDelftThe Netherlands
  2. 2.Mathematisch Instituut Universiteit LeidenLeidenThe Netherlands

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