Quasi-ergodicity of compact strong Feller semigroups on $L^2$
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866916619492524032 |
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| author | Kaleta, Kamil Schilling, René L. |
| author_facet | Kaleta, Kamil Schilling, René L. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_12834 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quasi-ergodicity of compact strong Feller semigroups on $L^2$ Kaleta, Kamil Schilling, René L. Functional Analysis Mathematical Physics Probability Spectral Theory Primary 47D06, Secondary 37A30, 47A35, 47D08, 60G53, 60J35 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. |
| title | Quasi-ergodicity of compact strong Feller semigroups on $L^2$ |
| topic | Functional Analysis Mathematical Physics Probability Spectral Theory Primary 47D06, Secondary 37A30, 47A35, 47D08, 60G53, 60J35 |
| url | https://arxiv.org/abs/2304.12834 |