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Sfb 288 Differential Geometry and Quantum Physics |
Abstract for Sfb Preprint No. 455
Long time behavior of one-dimensional stochastic dynamics Michael Eckhoff und Markus Klein
In this paper the authors analyze the long time behavior of certain Markov chains, namely jump processes of second order jump range, as the system size is growing. The motivation has its origin in statistical mechanics, where the time evolution of the magnetization in a Glauber dynamic of a mean field type spin system is considered. As a standard example might serve the Curie-Weiss model. The process considered can also be regarded as the space and time discrete analog of a one-dimensional randomly perturbed dynamical system. In leading order as the system size grows the authors derive in terms of the rate function of the reversible distribution transition probabilities and transition times describing metastability in this model. These quantities are characterized by second order difference equations for which the Green's function has a particular simple structure. This approach leads to the classical problem of the asymptotic behavior of sums of Laplace type.
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