What type of dynamics arise in E0-dilations of commuting quantum Markov processes?

Let H be a separable Hilbert space. Given two strongly commuting CP0-semigroups φ and θ on B(H), there is a Hilbert space K ⊇ H and two (strongly) commuting E0-semigroups α and β such that φs ◦ θt(PHAPH) = PHαs ◦ βt(A)PH for all s, t ≥ 0 and all A ∈ B(K). In this note we prove that if φ is not an automorphism semigroup then α is cocycle conjugate to the minimal ∗-endomorphic dilation of φ, and that if φ is an automorphism semigroup then α is also an automorphism semigroup. In particular, we conclude that if φ is not an automorphism semigroup and has a bounded generator (in particular, if H is finite dimensional) then α is a type I E0-semigroup.