Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 356 / 1 / b / Solution

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