 Lévy kernel of a Markov jump process

A kernel specifying the jump rate from state $x$ for each displacement $z$. With the displacement convention the infinitesimal generator is $L\phi (x)=\int (\phi (x+z)-\phi (x))K(x,dz)$. The destination convention instead records a rate measure on the new state, translating each displacement by $x$. For finite total rate, that total is the exponential holding rate and the normalized measure is the next-jump distribution.