  We consider stable solutions of semilinear equations in a very general
setting. The equation is set on a Polish topological space endowed with a
measure and the linear operator is induced by a carr\'e du champs
(equivalently, the equation is set in a diffusion Markov triple). Under
suitable curvature dimension conditions, we establish that stable solutions
with integrable carr\'e du champs are necessarily constant (weaker conditions
characterize the structure of the carr\'e du champs and carr\'e du champ
it\'er\'e). The proofs are based on a geometric Poincar\'e formula in this
setting. From the general theorems established, several previous results are
obtained as particular cases and new ones are provided as well.
