Two-parameter Hunt processes and potential theory
\textit{R. Z. Khas'minskij} [Trudy VI vsesojuzn. Soveščan. po Teor. Veroyatn. mat. Statist., Vil'njus-Palanga, 5-10, 12-14 sent. 1960, 177- 182 (1962; Zbl 0121.351)] and \textit{L. L. Helms} [J. Appl. Probab. 4, 130-136 (1967; Zbl 0314.60059)] have both treated the biharmonic equation \(\Delta^ 2u=0\) using one-parameter Brownian motion. The author treats this and similar equations using the idea of a two-parameter Markov process. Given two commuting one-parameter semigroups \(P^ 1_ s\) and \(P^ 2_ t\) with generators \({\mathcal L}_ 1\) and \({\mathcal L}_ 2\), he constructs a two-parameter semigroup \(P_{st}=P^ 1_ sP^ 2_ t=P^ 2_ tP^ 1_ s\). The semigroup leads to a two-parameter process \(X_{st}\) which enjoys certain Markov properties, and which is naturally associated with the operator \({\mathcal L}_ 1{\mathcal L}_ 2\). The author studies the potential theory associated with this process and uses it to give a probabilistic solution to a Dirichlet problem for the equation \({\mathcal L}_ 1{\mathcal L}_ 2=0\).
scientific article; zbMATH DE number 4115676