A characterization of Markov property for semigroups with invariant measure (Q1349064)

Let \((P_{t})\) be a continuous semigroup of operators on an \(L^{2}\) space. Suppose that the domain of its infinitesimal generator contains an algebra \(\mathcal {A}\). Then the ``carré du champ'' operator is defined as \(\Gamma (f, g):= {1 \over 2 }(L(f\cdot g) - f\cdot Lg- g\cdot Lf)\), for \(f, g \in \mathcal {A}\) [see \textit{J.-P. Roth}, C. R. Acad. Sci., Paris, Sér. A 278, 1103-1106 (1974; Zbl 0282.47010)]. The main result asserts that (under some mild hypotheses) \(\Gamma (f, f) \geq 0\), \(\forall f \in \mathcal {A}\), is equivalent to: \(P_{t} 1 = 1\) and \(f \geq 0\) \(\Rightarrow \) \(P_{t} f \geq 0\) \(\forall t > 0\), \(f \in L^{2}\). Some examples are finally considered.