A Brascamp-Lieb-Luttinger-type inequality and applications to symmetric stable processes (Q2723498)

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, NEWLINE\[NEWLINE \mathbb P_z(\tau_D>t)\leq \mathbb P_0(\tau_{D^*}>t) ,\quad t>0 , NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE R_D:=\sup\{R>0 : \exists \text{ball of radius} R \text{in} D\} NEWLINE\]NEWLINE and the corresponding symmetric strip NEWLINE\[NEWLINE S(D) := \{z=(z_1,\ldots,z_n) : -R_D<z_n<R_D\} . NEWLINE\]NEWLINE It has been proved that for unbounded convex \(D\)'s in \(\mathbb R^n\) [\textit{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 \textit{H. J. Brascamp, E. H. Lieb} and \textit{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 NEWLINE\[NEWLINE \mathbb P_z(\tau_{D,\alpha}>t)\leq \mathbb P_0(\tau_{S(D),\alpha}>t) ,\quad t\geq 0 , NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE \lambda_{D,\alpha} =\lim_{t\to\infty}\frac{1}{t}\log\mathbb P_z(\tau_{D,\alpha}>t).NEWLINE\]