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Exit times of symmetric stable processes from unbounded convex domains. (English) Zbl 1132.60071

The present work is concerned with asymptotic bounds on the tail of first exit time distributions of a symmetric \(\alpha\)-stable process to leave certain unbounded domains when starting inside. The domains \(D\) under investigation are convex generalizations of cones in the sense that there is a concave function \(\phi: \mathbb{R}_+\rightarrow\mathbb{R}_+\) with \(\lim_{x\rightarrow \infty}\frac{\phi(x)}{x} = 0\) and integrability conditions such that
\[ D=\{(x, y)\in \mathbb{R}\times \mathbb{R}^{d-1}: x>0 ~\text{and } ~\| y\| < \phi(x)\}. \]
In a first part the author developes estimates of the harmonic measure in \(\mathbb{R}^d\), i.e., the hitting probability to enter a certain set at the exit of \(D\). The author obtains lower and upper asymptotic estimates, stated in terms of the initial value \(z\in D\) and the geometry \(\phi\) and the growth of the in radios of the cross sections with the domain. In the sequel he applies these results to
\[ \phi(x) = x^\beta (\ln(1+x))^\mu, \qquad \beta \in [0,1], \;\mu\in\mathbb{R}, \]
and can give an example of an unbounded convex domain \(D\) with asymptotically subexponential behaviour of the first exit time distribution. According to the author this is an extension of results by Bañuelos and Bogdan.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J05 Discrete-time Markov processes on general state spaces
60J60 Diffusion processes
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