A spectral decomposition for the Bolthausen-Sznitman coalescent and the Kingman coalescent

Abstract

We consider both the Bolthausen-Sznitman and the Kingman coalescent restricted to the partitions of <span id="MathJax-Element-1-Frame" class="MathJax"><span id="MathJax-Span-1" class="math"><span id="MathJax-Span-2" class=""><span id="MathJax-Span-3" class="mo">{<span id="MathJax-Span-4" class="mn">1<span id="MathJax-Span-5" class="mo">,<span id="MathJax-Span-6" class="mo">…<span id="MathJax-Span-7" class="mo">,<span id="MathJax-Span-8" class="mi">n<span id="MathJax-Span-9" class="mo">}<span id="MathJax-Span-10" class="mo">. Spectral decompositions of the corresponding generators are derived. As an application we obtain a formula for the Green's functions and a short derivation of the well-known formula for the transition probabilities of the Bolthausen-Sznitman coalescent.