In this thesis we are concerned with the long time behavior of continuous time random walks on infinite graphs. The following three related problems are considered.

1. Stochastic completeness of the random walk. We characterize the stochastic completeness of the random walk in terms of function-theoretic and geometric properties of the underlying graph.

2. Uniqueness of the Cauchy problem for the discrete heat equation in certain function classes. We provide a uniqueness class on an arbitrary graph in terms of the growth of the L2-norm of solutions and show its sharpness. An application of this results to bounded solutions yields a criterion for stochastic completeness in terms of the volume growth with respect to a so-called adapted distance. In special cases, this leads to a volume growth criterion with respect to the graph distance as well.

3. Escape rate of the random walk. We provide upper rate functions for stochastically complete random walks in terms of the volume growth function.