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For reactions with stoichiometric vectors ${\nu }_r$ and reaction propensity functions $a_r$, the probability mass function $p(x,t)$ obeysIt is the forward operator of a Markov jump process equation for the copy-number continuous-time Markov chain.

For a function $h$ on the copy-number state space, define the shift operatorWith the propensities from part (a), the forward operator of a Markov jump process isEquivalently, the chemical master equation has the gain-minus-loss formwhere terms outside the state space vanish.

The adjoint Markov jump-process generator acts on observables:Indeed, shifting the summation index in the gain terms gives $⟨f,{\mathcal{L}}^{\ast }p⟩=⟨\mathcal{L}f,p⟩$.