ADMISSIBLE TRANSLATIONS OF THE BROWNIAN MOTION ON A LIE
GROUP
Abstract: The paper provides a new proof of Shigekawa’s theorem characterizing admissible
translations of the Wiener measure on a Lie group. We prove Shigekawa’s conditions to be
necessary finding the ”derivative” of the translation as a linear functional on a Hilbert space,
applying integrals of 1-forms along the paths of stochastic processes. We use the
classical Girsanov theorem as the main tool while obtaining the sufficiency in a
straightforward way. No advanced theorems concerning absolute continuity of measures
induced by stochastic processes are used, as was in Shigekawa’s original proof.
2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;
Key words and phrases: -