Infinitesimal generator of a Markov process

Summary

The infinitesimal generator of a Markov process is the operator that gives the rate of change of expected values at time $0$. It is the derivative at $t = 0$ of the transition semigroup of the process. For a Feller process the generator determines the law of the process. It links Markov processes with differential operators: Brownian motion has generator $\tfrac12 \Delta$.