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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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