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On Vervaat transform of Brownian bridges and Brownian motion

For a continuous function $f \in \mathcal{C}([0,1])$, define the Vervaat transform $V(f)(t):=f(τ(f)+t \mod1)+f(1)1_{\{t+τ(f) \geq 1\}}-f(τ(f))$, where $τ(f)$ corresponds to the first time at which the minimum of $f$ is attained. Motivated by recent study of quantile transforms for random walks and Brownian motion, we study the Vervaat transform of Brownian motion and Brownian bridges with arbitary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decompositions and study its semimartingale properties. The expectation and variance of the Vervaat transform of Brownian motion are also derived.