The Liouville theorem for a class of Fourier multipliers and its connection to coupling

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Abstract

The classical Liouville property says that all bounded harmonic functions in (Formula presented.), that is, all bounded functions satisfying (Formula presented.), are constant. In this paper, we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator (Formula presented.), such that the solutions (Formula presented.) to (Formula presented.) are Lebesgue a.e. constant (if (Formula presented.) is bounded) or coincide Lebesgue a.e. with a polynomial (if (Formula presented.) is polynomially bounded). 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 (Formula presented.) 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.

Details

Original languageEnglish
Pages (from-to)2374 - 2394
Number of pages21
JournalBulletin of the London Mathematical Society
Volume56
Issue number7
Publication statusPublished - Jul 2024
Peer-reviewedYes

External IDs

Scopus 85192534826

Keywords