The Liouville theorem for a class of Fourier multipliers and its connection to coupling
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916157906223104 |
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| author | Berger, David Schilling, René L. Shargorodsky, Eugene |
| author_facet | Berger, David Schilling, René L. Shargorodsky, Eugene |
| contents | The classical Liouville property says that all bounded harmonic functions in $\mathbb{R}^n$, i.e.\ all bounded functions satisfying $Δf = 0$, are constant. In this paper we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator $m(D)$, such that the solutions $f$ to $m(D)f=0$ are Lebesgue a.e.\ constant (if $f$ is bounded) or coincide Lebesgue a.e.\ with a polynomial (if $f$ grows like a polynomial). The class of Fourier multipliers includes the (in general non-local) generators of Lévy processes. For generators of Lévy processes we obtain necessary and sufficient conditions for a strong Liouville theorem where $f$ is positive and grows at most exponentially fast. As an application of our results above we prove a coupling result for space-time Lévy processes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_08929 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Liouville theorem for a class of Fourier multipliers and its connection to coupling Berger, David Schilling, René L. Shargorodsky, Eugene Probability Functional Analysis The classical Liouville property says that all bounded harmonic functions in $\mathbb{R}^n$, i.e.\ all bounded functions satisfying $Δf = 0$, are constant. In this paper we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator $m(D)$, such that the solutions $f$ to $m(D)f=0$ are Lebesgue a.e.\ constant (if $f$ is bounded) or coincide Lebesgue a.e.\ with a polynomial (if $f$ grows like a polynomial). The class of Fourier multipliers includes the (in general non-local) generators of Lévy processes. For generators of Lévy processes we obtain necessary and sufficient conditions for a strong Liouville theorem where $f$ is positive and grows at most exponentially fast. As an application of our results above we prove a coupling result for space-time Lévy processes. |
| title | The Liouville theorem for a class of Fourier multipliers and its connection to coupling |
| topic | Probability Functional Analysis |
| url | https://arxiv.org/abs/2211.08929 |