Fock Space Decomposition of Lévy Processes

We show that the general Levy process can be embedded in a suitable Fock space, classified by cocycles of the real line regarded as a group. The formula of de Finetti corresponds to coboundaries. Kolmogorov’s processes correspond to cocycles of which the derivative is a cocycle of the Lie algebra of R. The Lévy formula gives the general cocycle. 1 Cyclic representations of groups Let G be a group and g 7→ Ug a multiplier cyclic representation of G on a Hilbert space H, with multiplier σ : G×G→ C and cyclic vector Ψ. This means that • UgUh = σ(g, h)Ugh for all g, h ∈ G. • u(e) = I where e is the identity of then group and I is the identity operator on H. • Span {UgΨ : g ∈ G} is dense in H. If σ = 1 we say that U is a true representation. Recall that a multiplier of a group G is a measurable two-cocycle in Z2(G,U(1)); so σ is a map G×G→ U(1) such that σ(e, g) = σ(g, e) = 1 and σ(g, h)σ(g, hk)σ(gh, k)σ(h, k) = 1. (1) (1) expresses the associativity of operator multiplication of the U(g). σ is a coboundary if there is a map b : G→ U(1) with b(e) = 1 and σ(g, h) = b(gh)/(b(g)b(h)). (2) We also need the concept of a one cocycle ψ in a Hilbert space K carrying a unitary representation V . ψ is a map G→ K such that V (g)ψ(h) = ψ(gh) − ψ(g) for g, h ∈ G. (3) ψ is a coboundary if there is a vector ψ0 ∈ K such that ψ(g) = (V (g) − I)ψ0. (4) Coboundaries are always cocycles. We say that, in (2) and (4), σ is the coboundary of b and ψ is the coboundary of ψ0. We say that two cyclic σ-representations {H, U,Ψ} and {K, V,Φ} are cyclically equivalent if there exists a unitary operator W : H → K such that Vg = WUgW −1 for all g ∈ G, and WΨ = Φ. Any cyclic multiplier representation {H, U,Ψ} defines a function F on the group by F (g) := 〈Ψ, UgΨ〉, (5) which satisfies σ-positivity: