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.
Subject
Statistics and Probability
Reference6 articles.
1. Bakry D.,
Gentil I. and
Ledoux M., Analysis and geometry of Markov diffusion operators. Vol. 348 of
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].
Springer,
Cham
(2014).
2. Giannoulis J.,
Herrmann M. and
Mielke A.,
Continuum descriptions for the dynamics in discrete lattices: derivation and justification, in Analysis, Modeling and Simulation of Multiscale Problems.
Springer,
Berlin
(2006) 435–466.
3. Macià F.,
Propagación y control de vibraciones en medios discretos y continuos. Ph.D. thesis,
Universidad Complutense de Madrid, Departamento de Matemática Aplicada, Universidad Complutense de Madrid
(2002).
4. Wigner Measures in the Discrete Setting: High-Frequency Analysis of Sampling and Reconstruction Operators
5. Macroscopic Behavior of Microscopic Oscillations in Harmonic Lattices via Wigner-Husimi Transforms