Transition operator ($(Ph)(s)=\mathbb{E}[h(S_{t+1})\mid S_t=s]$)

The operator that averages a function over the next-state conditional law of a time-homogeneous Markov process. For a random walk with increments of law $\nu$ independent of the past including the initial state, $(Ph)(s)=\int h(s+y)\nu (dy)$. This additive operator preserves convexity when the integrals exist.