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QUASI-ERGODICITY OF COMPACT STRONG FELLER SEMIGROUPS ON
$L^2$
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu , First View
- Published online by Cambridge University Press:
- 14 November 2024, pp. 1-44
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We study the quasi-ergodicity of compact strong Feller semigroups
$U_t$, 
$t> 0$, on 
$L^2(M,\mu )$; we assume that M is a locally compact Polish space equipped with a locally finite Borel measue 
$\mu $. 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 
$\mu $ is infinite and the semigroup is not intrinsically ultracontractive. We relate quasi-ergodicity on 
$L^p(M,\mu )$ 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 
. 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.
. 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.