On the Convergence of Stochastic Process Convergence Proofs

Convergence of a stochastic process is an intrinsic property quite relevant for its successful practical for example for the function optimization problem. Lyapunov functions are widely used as tools to prove convergence of optimization procedures. However, identifying a Lyapunov function for a specific stochastic process is a difficult and creative task. This work aims to provide a geometric explanation to convergence results and to state and identify conditions for the convergence of not exclusively optimization methods but any stochastic process. Basically, we relate the expected directions set of a stochastic process with the half-space of a conservative vector field, concepts defined along the text. After some reasonable conditions, it is possible to assure convergence when the expected direction resembles enough to some vector field. We translate two existent and useful convergence results into convergence of processes that resemble to particular conservative vector fields. This geometric point of view could make it easier to identify Lyapunov functions for new stochastic processes which we would like to prove its convergence.