On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion

@article{Mielke2013OnTR,
  title={On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion},
  author={Alexander Mielke and Mark A. Peletier and D. R. Michiel Renger},
  journal={Potential Analysis},
  year={2013},
  volume={41},
  pages={1293-1327}
}
Motivated by the occurrence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions ℒ that induce a flow, given by ℒ(ρt,ρ̇t)=0$\mathcal{L} (\rho _{t},\dot \rho _{t})=0$. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be… 

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