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The Cauchy process and the Steklov problem. (English) Zbl 1055.60072
Given the Cauchy process \((X_{t})\) on \(\mathbb R^d\), \(d\geq 1\), the authors investigate the spectral properties of the semigroup \((P_{t}^D)\) obtained from \((X_{t})\) by killing the process upon leaving the bounded open set \(D\) whose boundary has to satisfy some (Lipschitz) regularity property. The basic step is the fact that the functions \(u_{n}(x,t):= P_{t}^D \varphi_{n}(x)\) satisfy a mixed Steklov problem where \(\varphi_{n}\) are the eigenfunctions of the semigroup \(P_{t}^D\) with eigenvalues \(e^{-\lambda_{n}t}\), \(0<\lambda_{1}<\lambda_{2}\leq \ldots\). Based on this and the explicitly known density \(p_{t}(x,y)\) of the transition function for the Cauchy process, variational formulas for \(\lambda_{n}\) are given as well as an estimate on the number of nodal parts of \(u_{n}\). Furthermore, the estimate \(\lambda_{n}\leq \sqrt{\mu_{n}}\) where \(\mu_{n}\) are the eigenvalues of the Dirichlet Laplacian on \(D\) is proven.
Another theme are the properties of the eigenfunctions: They are shown to be real analytic. Given a connected bounded “Lipschitz domain” being symmetric with respect to the \(x_{1}\)-axis the authors prove the existence of an eigenfunction \(\varphi_{\ast}\) antisymmetric relative to the \(x_{1}\)-axis with \(\pm \varphi_{\ast}(x) >0\) on \(D_{\pm} = \{x\in D\;| \;\pm x_{1} >0\}\) and show that \(\varphi_{\ast}\) has the lowest eigenvalue among all antisymmetric eigenfunctions. The last section is devoted to the one-dimensional problem with \(D=(-1,1)\). Estimates for \(\lambda_{1},\;\lambda_{2}\) and \(\lambda_{3}\) are given and symmetry properties of the corresponding eigenfunctions are proven.

MSC:
60J45 Probabilistic potential theory
60G52 Stable stochastic processes
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