On Nonexistence of any λ0-invariand positive harmonic function, a counter example to stroock' conjecture
Consider a sub-Markovian semigroup such that λ0, the border number between recrrence and transience, equals zero. In 1982, D. W. Stroock conjectured that under general hypotheses on the semi-group the corresponding process always admits an invariant measure. In this paper we present an example of a second order elliptic operator P with a generalized principal eigenvalue λ0 which equals zero such that the parabolic equation does not admit any positive invariant P—harmonic function and also any invariant measure. This gives a counter example to Stroock's conjecture for diffusion processes. We also present an example of a complete Riemannian manifold M which does not admit any positive invariant harmonic function while λ0, the bottom of the spectrum of M, is zero. This gives a partial answer to a question of Stroock and Sullivan.