Lp -independence of spectral bounds of non-local Feynman-Kac semigroups

Abstract

For a symmetric stable process we consider a transform of its transition semigroup by a non-local Feynman-Kac functional. We prove that if the Feynman-Kac functional belongs to a certain class, then the spectral bound of the transformed semigroup on L p (R d ) is independent of p .2000 Mathematics Subject Classification: 60J45, 60J40, 35J10.

Lp-independence of spectral bounds of Feynman–Kac semigroups by continuous additive functionals

“…In Takeda [27], he established the necessary and sufficient condition for the L p -independence of the spectral bounds for Feynman-Kac semigroup by continuous additive functionals having bounded variation whose Revuz measures are of Kato class having 0-order Green-tightness in the framework of symmetric conservative Markov processes under some conditions. The method of [27] also depends on the Donsker-Varadhan's large deviation theory and remains valid for non-local Feynman-Kac semigroup (see Kim [14], Takeda and Tawara [29], Tawara [31,32]). They assumed the transience of the underlying process for the notion of 0-order Green-tight measures of Kato class.…”

“…But, by imposing stronger assumptions on measures related to our Feynman-Kac semigroup, we can eliminate the Feller property of the underlying process for Theorems 1.1 and 1.2(1) (Remarks 3.1 and 4.1), which covers one of the results in [28]. In order to deduce our results, we also use the Donsker-Varadhan's large deviation theory as in [25][26][27]29,31]. Our conditions on measures are milder than theirs because of the refinement of the condition for doubly Feller property of the semigroups (see [6]) and the recent developments of the Feynman-Kac formula by continuous additive functionals of zero energy (see [8,4,5]).…”

Large deviations for discontinuous additive functionals of symmetric stable processes

“…Takeda and Tawara [28] and Tawara [31] showed the L p -independence of spectral bounds of Schrödinger type operators.…”

“…We showed in [28,Corollary 3.9] that the L p -independence of spectral bounds implies that the limit…”

“…For the proof of the L p -independence, they used heat kernel estimates. In [26], [28], [31], we proved by a completely different method that if λ 2 (θF ) ≤ 0, then λ p (θF ) = λ 2 (θF ) for 1 ≤ p ≤ ∞; we used again arguments in Donsker-Varadhan's large deviation theory. In particular, we investigate the contribution of the infinity to the so-called I-function.…”