Potential Analysis

, Volume 28, Issue 1, pp 17–33 | Cite as

Differentiability of Spectral Functions for Relativistic α-Stable Processes with Application to Large Deviations

  • Kaneharu Tsuchida


We consider the relativistic α-stable process, a pure jump Markov process generated by \(\mathcal{H}^{\alpha} = (-\Delta + m^{2/\alpha})^{\alpha /2}-m\). Let −C(λ) be the bottom of spectrum of Schrödinger type operator \(\mathcal{H}^{\lambda \mu} = \mathcal{H}^{\alpha} - \lambda \mu\), where μ is a signed Kato measure. We prove the differentiability of C(λ). As an application of it, we establish a large deviation principle for the additive functional \(A_t^{\mu}\) corresponding to the measure μ.


Symmetric relativistic stable process Spectral function Criticality Large deviation principle Kato class 

Mathematics Subject Classifications (2000)

60J45 60J55 35J10 


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© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Mathematical InstituteTohoku UniversitySendaiJapan

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