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Some geometric aspects of potential theory. (English) Zbl 0558.60055
Stochastic analysis and applications, Proc. int. Conf., Swansea 1983, Lect. Notes Math. 1095, 130-154 (1984).
[For the entire collection see Zbl 0543.00010.]
Given a Lévy process \(X=(X_ t)\) with distributions \((\mu_ t)_{t\geq 0}\) on the Euclidean space \({\mathbb{R}}^ d\). The aim of the paper is to give a criterion characterizing the class of essentially polar sets for X and then use it to interpret some known potential theoretic aspects of Lévy processes which are essentially due to the translation invariant property of the semigroup operators (the author called this the geometric structure of Lévy processes!).
Let \(T_ k\) be the hitting-time of X for the compact set \(k\subset {\mathbb{R}}^ d\), denote by \(\phi^{\lambda}(x):=E^ x(e^{-\lambda T_ k};\quad T_ k<+\infty)\) then k is said to be polar (resp. essentially polar) if, \(\phi^{\lambda}(x)=0\) (resp. \(\phi^{\lambda}(x)=0\) \(\Lambda\).a.S) for \(\forall x\in {\mathbb{R}}^ d\) (resp. where \(\Lambda\) (dx) is the Lebesgue measure in \({\mathbb{R}}^ d)\). It is proved that k is not essentially polar if and only if there is a Radon measure \(\mu\) on k such that, for some \(\lambda >0\), \[ J^{\lambda}(\mu)=(2\pi)^{-d}\int Re(1/(\lambda +\psi (u)))| {\hat \mu}(z)|^ 2dz<+\infty, \] where \({\hat \mu}\)(z) is the Fourier transform of \(\mu\) and where \(e^{- t\psi}(u)\) is the characteristic function of \(X_ t-X_ 0\).
Reviewer: X.L.Nguyen

60J45 Probabilistic potential theory
60E10 Characteristic functions; other transforms