Forward operator of a Markov jump process ($L^{\ast }$)

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 $Lf(x)={\sum }_r{a}_r(x)[f(x+{\nu }_r)-f(x)]$.