This text is a self-contained account of Cramer's theory in infinite dimensions. The point of view is slightly different from the classical texts of Azencott, Bahadur and Zabell, Dembo and Zeitouni, and Deuschel and Stroock. The authors have been trying to understand the relevance of the topological hypotheses necessary to carry out the core of the theory. They have also drawn some inspiration from the analogy between the large deviation proofs in statistical mechanics and for i.i.d. random variables.

On Cramers Theory in Infinite Dimensions

Cerf, Raphaël

Societe Mathematique De France

This monograph studies a series of mathematical models of the evolution of a population under mutation and selection. Its starting point is the quasispecies equation, a general non-linear equation which describes the mutation-selection equilibrium in Manfred Eigen’s famous quasispecies model. A detailed analysis of this equation is given under the assumptions of finite genotype space, sharp peak landscape, and class-dependent fitness landscapes. Different probabilistic representation formulae are derived for its solution, involving classical combinatorial quantities like Stirling and Euler numbers. 
It is shown how quasispecies and error threshold phenomena emerge in finite population models, and full mathematical proofs are provided in the case of the Wright–Fisher model. Along the way, exact formulas are obtained for the quasispecies distribution in the long chain regime, on the sharp peak landscape and on class-dependent fitness landscapes. 
Finally, several other classical population models are analyzed, with a focus on their dynamical behavior and their links to the quasispecies equation. 
This book will be of interest to mathematicians and theoretical ecologists/biologists working with finite population models.

The Quasispecies Equation and Classical Population Models (Probability Theory and Stochastic Modelling, 102)

Science & Math

Dalmau, Joseba

Biological Sciences

Springer

The Quasispecies Equation and Classical Population Models (Probability Theory and Stochastic Modelling)

This volume is a synopsis of recent works aiming at a mathematically rigorous justification of the phase coexistence phenomenon, starting from a microscopic model. It is intended to be self-contained. Those proofs that can be found only in research papers have been included, whereas results for which the proofs can be found in classical textbooks are only quoted.

The Wulff Crystal in Ising and Percolation Models: Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004 (Lecture Notes in Mathematics, 1878)

Mathematics

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