The Eyring-Kramers law for Markovian jump processes with symmetries

Nils Berglund (1, 2) , Sébastien Dutercq (1, 2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 FDP - Fédération de recherche Denis Poisson

We prove an Eyring-Kramers law for the small eigenvalues and mean first-passage times of a metastable Markovian jump process which is invariant under a group of symmetries. Our results show that the usual Eyring-Kramers law for asymmetric processes has to be corrected by a factor computable in terms of stabilisers of group orbits. Furthermore, the symmetry can produce additional Arrhenius exponents and modify the spectral gap. The results are based on representation theory of finite groups.

Mots clés: symmetry group; spectral theory; Markovian jump process; Metastability; Kramers' law; stochastic exit problem; first-passage time; representation theory

Domaines: Probabilités [math.PR]; Théorie des représentations [math.RT]

Nils Berglund, Sébastien Dutercq. The Eyring-Kramers law for Markovian jump processes with symmetries. Journal of Theoretical Probability, 2015, 29 (4), pp.1240-1279. ⟨10.1007/s10959-015-0617-9⟩. ⟨hal-00913302v2⟩