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The Hausdorff lower semicontinuous envelope of the length in the plane

Raphaël Cerf — 2002

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We study the Hausdorff lower semicontinuous envelope of the length in the plane. This envelope is taken with respect to the Hausdorff metric on the space of the continua. The resulting quantity appeared naturally as the rate function of a large deviation principle in a statistical mechanics context and seems to deserve further analysis. We provide basic simple results which parallel those available for the perimeter of Caccioppoli and De Giorgi.

The exit path of a Markov chain with rare transitions

Olivier CatoniRaphaël Cerf — 2010

ESAIM: Probability and Statistics

We study the exit path from a general domain after the last visit to a set of a Markov chain with rare transitions. We prove several large deviation principles for the law of the succession of the cycles visited by the process (the cycle path), the succession of the saddle points gone through to jump from cycle to cycle on the cycle path (the saddle path) and the succession of all the points gone through (the exit path). We estimate the time the process spends in each cycle of the cycle path...

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