On stochastic completeness of weighted graphs [Elektronische Ressource] / Xueping Huang. Fakultät für Mathematik. SFB 701 Spectral Structures and Topological Methods in Mathematics

On stochastic completeness of weightedgraphsXueping HuangA Dissertation Submitted for the Degree of Doctoratthe Department of Mathematics Bielefeld University4 June 2011On stochastic completeness of weightedgraphsDissertation zur Erlangung des Doktorgradesder Fakulta¨t fu¨r Mathematikder Universit¨at Bielefeldvorgelegt vonXueping Huang4 June 2011Gedruckt auf alterungsbest¨andigem Papier nach DIN–ISO 9706AbstractIn this thesis we are concerned with the long time behavior of continuous time ran-dom walks on infinite graphs. The following three related problems are considered.1. Stochastic completeness of the random walk. We characterize the stochas-tic completeness of the random walk in terms of function-theoretic and geometricproperties of the underlying graph.2. Uniqueness of the Cauchy problem for the discrete heat equation in certainfunction classes. We provide a uniqueness class on an arbitrary graph in terms of2the growth of the L -norm of solutions and show its sharpness. An application ofthis results to bounded solutions yields a criterion for stochastic completeness interms of the volume growth with respect to a so-called adapted distance. In specialcases, this leads to a volume growth criterion with respect to the graph distance aswell.3. Escape rateoftherandom walk. Weprovide upper ratefunctions forstochas-tically complete random walks in terms of the volume growth function.