Excessive function

Summary

An excessive function of a Markov process is a nonnegative function that decreases on average along the paths of the process. It is also recovered as the limit of its averages at small times. Excessive functions are the probabilistic counterpart of nonnegative superharmonic functions. For Brownian motion the two classes coincide. For general Markov processes, excessive functions play the role of superharmonic functions in an abstract potential theory.