Markov jump-process generator

For transition rates $q_{nm}$, the generator acts on a function $f$ byWhenever the expectations are finite,Choosing polynomial functions $f$ gives differential equations for the moments.

For jumps $x\mapsto x+{\nu }_r$ of rates $a_r(x)$, the forward operator acts on a probability mass function byIt is adjoint to the Markov jump-process generator $Lf(x)={\sum }_r{a}_r(x)[f(x+{\nu }_r)-f(x)]$.