Complex intertwinings and quantification of discrete free motions
ESAIM: Probability and Statistics, Tome 23 (2019), pp. 409-429

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$$, 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.

Reçu le :
Accepté le :
DOI : 10.1051/ps/2018020
Classification : 81Q35, 47D08, 35K08, 39A12, 60J27
Keywords: Quantification, free motion, Markov process, Doob transform, intertwining

Miclo, Laurent 1

1 
@article{PS_2019__23__409_0,
     author = {Miclo, Laurent},
     title = {Complex intertwinings and quantification of discrete free motions},
     journal = {ESAIM: Probability and Statistics},
     pages = {409--429},
     year = {2019},
     publisher = {EDP Sciences},
     volume = {23},
     doi = {10.1051/ps/2018020},
     zbl = {1420.81011},
     mrnumber = {3980426},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps/2018020/}
}
TY  - JOUR
AU  - Miclo, Laurent
TI  - Complex intertwinings and quantification of discrete free motions
JO  - ESAIM: Probability and Statistics
PY  - 2019
SP  - 409
EP  - 429
VL  - 23
PB  - EDP Sciences
UR  - https://www.numdam.org/articles/10.1051/ps/2018020/
DO  - 10.1051/ps/2018020
LA  - en
ID  - PS_2019__23__409_0
ER  - 
%0 Journal Article
%A Miclo, Laurent
%T Complex intertwinings and quantification of discrete free motions
%J ESAIM: Probability and Statistics
%D 2019
%P 409-429
%V 23
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/ps/2018020/
%R 10.1051/ps/2018020
%G en
%F PS_2019__23__409_0
Miclo, Laurent. Complex intertwinings and quantification of discrete free motions. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 409-429. doi: 10.1051/ps/2018020

[1] D. Bakry, I. Gentil and M. Ledoux, Analysis and geometry of Markov diffusion operators. Vol. 348 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Cham (2014). | Zbl | MR | DOI

[2] J. Giannoulis, M. Herrmann and A. Mielke, Continuum descriptions for the dynamics in discrete lattices: derivation and justification, in Analysis, Modeling and Simulation of Multiscale Problems. Springer, Berlin (2006) 435–466. | MR

[3] F. Macià, 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] F. Macià, Wigner measures in the discrete setting: high-frequency analysis of sampling and reconstruction operators. SIAM J. Math. Anal. 36 (2004) 347–383. | Zbl | MR | DOI

[5] A. Mielke, Macroscopic behavior of microscopic oscillations in harmonic lattices via Wigner-Husimi transforms. Arch. Ration. Mech. Anal. 181 (2006) 401–448. | Zbl | MR | DOI

[6] M. Zworski, Semiclassical Analysis. Vol. 138 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2012). | Zbl | MR

Cité par Sources :