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Let ${\mathbf{e}}_i$ be the $i$th coordinate vector and take $p$ to be zero whenever any argument is negative. Right jumps $i\to i+1$, left jumps $i\to i-1$, and absorption from compartment $1$ give the chemical master equation

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 $⟨f,{\mathcal{L}}^{\ast }p⟩=⟨\mathcal{L}f,p⟩$.