Abstract
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: ℤm, 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.
Highlights
Speaking, quantification is the mathematical link between classical and quantum mechanics and has led to the tremendous development of semi-classical limits, see e.g. the book of Zworski [6] and the references therein
It leads us to propose a general definition of quantification in the context of Markov process theory and to see that it is meaningful, at least for the simplest examples of free motions
A similar result holds on general state spaces V and Markov generators L, by considering a family pFhqhą0 such that |Fhpt, y, 0; xq|2 μpdxq is an approximation of the Dirac mass at y and Fhpt, y, 0; xq is symmetrical in y, x and does not depend on t
Summary
Quantification is the mathematical link between classical and quantum mechanics and has led to the tremendous development of semi-classical limits, see e.g. the book of Zworski [6] and the references therein. We would like to consider fixed discrete state spaces, even finite sets, e.g. just on Z3 := Z{p3Zq. It leads us to propose a general definition of quantification in the context of Markov process theory and to see that it is meaningful, at least for the simplest examples of free motions. A similar result holds on general state spaces V and Markov generators L, by considering a family pFhqhą0 such that |Fhpt, y, 0; xq|2 μpdxq is an approximation of the Dirac mass at y and Fhpt, y, 0; xq is symmetrical in y, x and does not depend on t This kind of degeneracy could be avoided by requiring in (H3) that ξ0 P Tzt1, 1u, the case ξ “ ́1 appearing for instance when L generates deterministic motions that can be reversed in time. Hamiltonian dynamics on graphs whose quantification corresponds to Metropolis algorithms is an interesting perspective in the field of optimizing and sampling stochastic algorithms
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