Quasi-ergodicity of compact strong Feller semigroups on $L^2$
Kamil Kaleta, René L. Schilling
TL;DR
The paper develops a unified theory for quasi-ergodicity of compact strong Feller semigroups on L^2(M,μ), even when μ is infinite and the semigroups are non-conservative. By linking quasi-ergodicity to heat content via Z(t) and to ground-state domination (aGSD and pGSD), it derives exponential and progressive uniform convergence bounds toward a unique quasi-stationary measure m (and its adjoint m^*), expressed through the ground states φ_0, ψ_0 and eigenvalue λ_0. It introduces progressive ground state domination (pGSD) as a central mechanism that yields progressive uniform quasi-ergodicity (pUQE) and provides necessary and sufficient conditions in broad settings, including Feynman–Kac semigroups and non-local Schrödinger operators. The results encompass large-time asymptotics, explicit space–time rates, and a wide range of applications (Lévy, Lévy-type processes, Markov chains, fractals, and the harmonic oscillator), offering sharp two-sided heat kernel insights and understanding of long-time behavior in non-self-adjoint, non-compact contexts.
Abstract
We study the quasi-ergodicity of compact strong Feller semigroups $U_t$, $t > 0$, on $L^2(M,μ)$; we assume that $M$ is a locally compact Polish space equipped with a locally finite Borel measue $μ$. The operators $U_t$ are ultracontractive and positivity preserving, but not necessarily self-adjoint or normal. We are mainly interested in those cases where the measure $μ$ is infinite and the semigroup is not intrinsically ultracontractive. We relate quasi-ergodicity on $L^p(M,μ)$ and uniqueness of the quasi-stationary measure with the finiteness of the heat content of the semigroup (for large values of $t$) and with the progressive uniform ground state domination property. The latter property is equivalent to a variant of quasi-ergodicity which progressively propagates in space as $t \uparrow \infty$; the propagation rate is determined by the decay of $U_t \mathbb{1}_M(x)$. We discuss several applications and illustrate our results with examples. This includes a complete description of quasi-ergodicity for a large class of semigroups corresponding to non-local Schrödinger operators with confining potentials.