Large Deviations for Stochastic Processes

Stefan Adams

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Submission date: 07. Jan. 2013
Pages: 31
Keywords and phrases: large deviation principle, Fleming's semigroup, viscosity solution, range condition
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Abstract:
The notes are devoted to results on large deviations for sequences of Markov processes following closely the book by Feng and Kurtz (J. Feng and T.G. Kurtz, Large Deviations for Stochastic Processes, American Mathematical Society (2006)). We outline how convergence of Fleming's nonlinear semigroups (logarithmically transformed nonlinear semigroups) implies large deviation principles analogous to the use of convergence of linear semigroups in weak convergence. The latter method is based on the range condition for the corresponding generator. Viscosity solution methods however provide applicable conditions for the necessary nonlinear semigroup convergence. Once having established the validity of the large deviation principle one is concerned to obtain more tractable representations for the corresponding rate function. This in turn can be achieved once a variational representation of the limiting generator of Fleming's semigroup can be established. The obtained variational representation of the generator allows for a suitable control representation of the rate function. The notes conclude with a couple of examples to show how the methodology via Fleming's semigroups works. These notes are based on the mini-course 'Large deviations for stochastic processes' the author held during the workshop 'Dynamical Gibs-non-Gibbs transitions' at EURANDOM in Eindhoven, December 2011, and at the Max-Planck Institute for Mathematics in the Sciences in Leipzig, July 2012.