The Segal–Bargmann Transform for Lévy Functionals

Let S′ be the space of tempered distributions on R and Λ the Lévy noise measure on S′. In this paper, we derive the closed form of the Segal–Bargmann transform (or the S-transform) of the Lévy functionals on L2(S′, Λ) and show that S-transform is a unitary operator from L2(S′, Λ) onto the space of Bargmann–Segal analytic functions on L2(R2, λ), where dλ=dt⊕u2 dβ0(u) and β0 is the Lévy measure.