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A Brascamp-Lieb-Luttinger-type inequality and applications to symmetric stable processes. (English) Zbl 0974.60037

Let \(D\subseteq \mathbb R^n\) be a domain and let \(D^*\) be a ball in \(\mathbb R^n\), centered at zero, possessing the same volume as \(D\). Several classical results relate quantities of \(D\) with those of \(D^*\). For example, if \(\lambda_D\) (or \(\lambda_{D^*}\)) denotes the first eigenvalue of the Dirichlet Laplacian in \(D\) (or \(D^*\)), then \( \lambda_{D^*}\leq \lambda_D.\) Similarly, \[ \mathbb P_z(\tau_D>t)\leq \mathbb P_0(\tau_{D^*}>t) ,\quad t>0 , \] where \(\tau_D\) (or \(\tau_{D^*}\)) denotes the first exit time out of \(D\) of an \(n\)-dimensional Brownian motion started in \(z\in D\) (or out of \(D^*\) started at zero). The purpose of the present paper is to treat those questions for unbounded domains \(D\). Of course, then \(D^*\) makes no longer sense. Instead one investigates the inradius \(R_D\) of \(D\) defined by \[ R_D:=\sup\{R>0 : \exists \text{ball of radius} R \text{in} D\} \] and the corresponding symmetric strip \[ S(D) := \{z=(z_1,\ldots,z_n) : -R_D<z_n<R_D\} . \] It has been proved that for unbounded convex \(D\)’s in \(\mathbb R^n\) [C. Bandle, “Isoperimetric inequalities and applications” (1980; Zbl 0436.35063)] inequality holds in the form \(\lambda_{S(D)}\leq \lambda_D.\) A first result in this paper relates certain integrals over \(D\) to those over \(S(D)\). In the case of bounded \(D\)’s those estimates were proved by H. J. Brascamp, E. H. Lieb and J. M. Luttinger [J. Funct. Anal. 17, 227-237 (1974; Zbl 0286.26005)]. An interesting application is as follows: Let \(X^\alpha:=(X_t^\alpha)_{t\geq 0}\) be a \(2\)-dimensional \(\alpha\)-Lévy motion for a certain \(\alpha\in(0,2]\) and let \(\tau_{D,\alpha}\) be the first exit time of \(X^\alpha\) out of the convex domain \(D\subseteq \mathbb R^2\). Then it follows that \[ \mathbb P_z(\tau_{D,\alpha}>t)\leq \mathbb P_0(\tau_{S(D),\alpha}>t) ,\quad t\geq 0 , \] where as before the process on the left-hand side is started at \(z\in D\). As a consequence, estimates for the first eigenvalues \(\lambda_{D,\alpha}\) of generalized Laplacian over \(D\) are derived. Note that the exit time and this Laplacian are connected via \[ \lambda_{D,\alpha} =\lim_{t\to\infty}\frac{1}{t}\log\mathbb P_z(\tau_{D,\alpha}>t). \]

MSC:

60G52 Stable stochastic processes
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
60J45 Probabilistic potential theory
35J99 Elliptic equations and elliptic systems
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