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alpha-Wiener bridges: singularity of induced measures and sample path properties

Let us consider the process $(X_t^{(α)})_{t\in[0,T)}$ given by the SDE $dX_t^{(α)} = -\fracα{T-t}X_t^{(α)} dt+ dB_t$, $t\in[0,T)$, where $α\in R$, $T\in(0,\infty)$, and $(B_t)_{t\geq 0}$ is a standard Wiener process. In case of $α>0$ the process $X^{(α)}$ is known as an $α$-Wiener bridge, in case of $α=1$ as the usual Wiener bridge. We prove that for all $α,β\in R$, $α\neβ$, the probability measures induced by the processes $X^{(α)}$ and $X^{(β)}$ are singular on C[0,T). Further, we investigate regularity properties of $X_t^{(α)}$ as $t\uparrow T$.