Functional inequalities for time-changed symmetric α-stable processes

Jian WANG; Longteng ZHANG

doi:10.1007/s11464-021-0908-7

Front. Math. China ›› 2021, Vol. 16 ›› Issue (2) : 595 -622. DOI: 10.1007/s11464-021-0908-7

RESEARCH ARTICLE

Functional inequalities for time-changed symmetric $α$ -stable processes

Jian WANG ¹^,^†
, Longteng ZHANG ²

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Abstract

We establish sharp functional inequalities for time-changed symmetric $α$ -stable processes on $ℝ d$ with $d ≥ 1$ and $α ∈ (0, 2)$ , which yield explicit criteria for the compactness of the associated semigroups. Furthermore, when the time change is defined via the special function $W (x) = (1 + | x |) β$ with $β > α$ we obtain optimal Nash-type inequalities, which in turn give us optimal upper bounds for the density function of the associated semigroups.

Keywords

$α$ -stable process')">Symmetric $α$ -stable process / time change, functional inequality

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Jian WANG, Longteng ZHANG. Functional inequalities for time-changed symmetric

α

-stable processes. Front. Math. China, 2021, 16(2): 595-622 DOI:10.1007/s11464-021-0908-7

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