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Green Function for an Asymptotically Stable Random Walk in a Half Space

Autor(en)
Denisov, Denis ; Wachtel, Vitali
Normierter Zeitschriftentitel
Journal of theoretical probability 
DOI
10.1007/s10959-023-01283-4
URL
https://pub.uni-bielefeld.de/record/2984081
Web of Science-ID
001074926500001

Weitere Details

Sprache
englisch
Darstellungsform
Text
Schlagwörter
Conditioned process , Exit time , Harmonic function , Random walk

We consider an asymptotically stable multidimensional random walk S(n) = ( S-1(n),..., S-d (n)). For every vector x = (x(1)..., x(d)) with x(1) >= 0, let tau(x) := min{n > 0 : x(1) + S-1(n) <= 0} be the first time the random walk x + S(n) leaves the upper half space. We obtain the asymptotics of p(n)(x, y) := P( x + S(n) epsilon y + Delta, tau(x) > n) as n tends to infinity, where Delta is a fixed cube. From that, we obtain the local asymptotics for the Green function G(x, y) := Sigma(n) pn(x, y), as | y| and/or |x| tend to infinity.

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