Transition operator
= Transition operator
{title2=$(Ph)(s)=\mathbb E[h(S_{t+1})\mid S_t=s]$}
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.