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

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Let us consider the process (Xₜ⁽α⁾)ₜ∈[₀,ₜ₎ given by the SDE dXₜ⁽α⁾ = -αT-tXₜ⁽α⁾ dt+ dBₜ, t∈[0,T), where α∈ R, T∈(0,∞), and (Bₜ)ₜ≥ 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∈ R, α≠β, the probability measures induced by the processes X⁽α⁾ and X⁽β⁾ are singular on C[0,T). Further, we investigate regularity properties of Xₜ⁽α⁾ as t T.