Exponential decay of positivity preserving semigroups on \(L^p\) (Q1591239)

Under the the hyperboundedness-like conditions (I) [by \textit{L. Gross}, in J. Funct. Anal. 10, 52-109 (1972; Zbl 0237.47012)], the existence of invariant measures for given Markovian (or positive preserving) semigroups (including the infinite-dimensional cases) are proved. In order to consider the exponential dacay of the Markov semigroup \((P_t)\) in infinite dimension to a project operator, an ergodic condition (E) [given by \textit{S. Kusuoka}, ibid. 103, No. 2, 229-274 (1992; Zbl 0772.58003)] is introduced as a substitution for the strict positivity of transition densities. It is proved that \((P_t)\) is exponential dacay to a project operator in \(L_p\) sense iff conditions (I) and (E) hold.