A remark on quasi-ergodicity of ultracontractive Markov processes

In this note we show that for a killed Markov process determined by a certain selfadjoint operator on a bounded domain, there are two quite different quasi-ergodic behaviors, corresponding to the Yaglom limit and a certain “fractional” Yaglom limit respectively. We also show that the high (higher than 2) order Dirichlet eigenvalues of the operator can be used to characterize the exponential convergence rate to quasi-stationarity for some marginals of the Markov process conditioned not to exit a bounded domain. These convergence rates depend on the starting points.