A Variant of Pitman’s Theorem on $$(2J_s-R_s,s\ge 0)$$ ( 2 J s - R s , s ≥ 0 ) for a General Transient Bessel Process $$R_{(+)}$$ R ( + ) and Its Implications for the Corresponding Ito’s Measure $$\large \mathbf{n}_{(-)}$$ n ( - )

Projection properties of the future infimum of a transient Bessel process $$R_{(+)}$$ R ( + ) with dimension $$d_{(+)}=2(1+\alpha )$$ d ( + ) = 2 ( 1 + α ) $$(\alpha \in (0,1))$$ ( α ∈ ( 0 , 1 ) ) as well as the definition of the local time $$(L_t)$$ ( L t ) of a recurrent Bessel process $$R_{(-)}$$ R ( - ) with dimension $$d_{(-)}=2(1-\alpha )$$ d ( - ) = 2 ( 1 - α ) as the compensator of $$(R_{(-)})^{2\alpha }$$ ( R ( - ) ) 2 α may be seen to play some hidden but quite efficient role to obtain several integral representation formulae for the excursion theory of the $$R_{(-)}$$ R ( - ) process away from 0. The precise formulae, which involve simple universal constants, are quite useful when dealing with the whole family of Bessel processes for dimensions between 0 and 2 (i.e., the reflecting case) and between 2 and 4 (i.e., the transient case).