Some distributional properties of a Brownian motion with a drift and an extension of P. Lévy's theorem (Q2711133)

The present paper starts by explaining connections between various known generalizations of Lévy's theorem \((\sup B-B, \sup B) = (|B|, L(B))\) in law, where \(L(B)\) is the local time at \(0\), from a standard Brownian motion \(B\) to Brownian motion with drift. Then new generalizations to Brownian motion with {random} drift and a class of conditionally Gaussian martingales are given. This type of results associate to the original process \(B\) a partner process \(X\) such that \((\sup B-B, \sup B)= (|X|, L(X))\) in law. As in the most popular proof of Lévy's original result, the new proofs are based on a lemma of Skorokhod.