@article{BaakeBaakeBovieretal.2005,
  author    = {Baake, Ellen and Baake, Michael and Bovier, Anton and Klein, Markus},
  title     = {An asymptotic maximum principle for essentially linear evolution models},
  issn      = {0303-6812},
  year      = {2005},
  abstract  = {Recent work on mutation-selection models has revealed that, under specific assumptions on the fitness function and the mutation rates, asymptotic estimates for the leading eigenvalue of the mutation-reproduction matrix may be obtained through a low-dimensional maximum principle in the limit N --> infinity (where N, or N-d with d greater than or equal to 1, is proportional to the number of types). In order to extend this variational principle to a larger class of models, we consider here a family of reversible matrices of asymptotic dimension N-d and identify conditions under which the high-dimensional Rayleigh-Ritz variational problem may be reduced to a low-dimensional one that yields the leading eigenvalue up to an error term of order 1/N. For a large class of mutation-selection models, this implies estimates for the mean fitness, as well as a concentration result for the ancestral distribution of types},
  language  = {en}
}
@article{BovierGayrardKlein2005,
  author    = {Bovier, Anton and Gayrard, Veronique and Klein, Markus},
  title     = {Metastability in reversible diffusion processes : II. Precise asymptotics for small eigenvalues},
  issn      = {1435-9855},
  year      = {2005},
  abstract  = {We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form -epsilonDelta + delF(.) del on R-d or subsets of Rd, where F is a smooth function with finitely many local minima. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of the spectrum is given, up to multiplicative errors tending to one, by the eigenvalues of the classical capacity matrix of the array of capacitors made of balls of radius epsilon centered at the positions of the local minima of F. We also get very precise uniform control on the corresponding eigenfunctions. Moreover, these eigenvalues can be identified with the same precision with the inverse mean metastable exit times from each minimum. In [BEGK3] it was proven that these mean times are given, again up to multiplicative errors that tend to one, by the classical Eyring- Kramers formula},
  language  = {en}
}