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The traditional quantification of free motions on Euclidean spaces into the Laplacian is revisited as a complex intertwining obtained through Doob transforms with respect to complex eigenvectors. This approach can be applied to free motions on finitely generated discrete Abelian groups: ℤ  m  , with  m  ∈ ℕ, finite tori and their products. It leads to a proposition of Markov quantification. It is a first attempt to give a probability-oriented interpretation of exp( ξL ), when  L  is a (finite) Markov generator and  ξ  is a complex number of modulus 1.

Complex intertwinings and quantification of discrete free motions