Spectral theory of discrete processes

• Authors: Palle Jorgensen , Myung-Sin Song
• Languages of publication: EN

Abstracts

EN

We offer a spectral analysis for a class of transfer operators. These transfer operators arise for a wide range of stochastic processes, ranging from random walks on infinite graphs to the processes that govern signals and recursive wavelet algorithms; even spectral theory for fractal measures. In each case, there is an associated class of harmonic functions which we study. And in addition, we study three questions in depthIn specific applications, and for a specific stochastic process, how do we realize the transfer operator T as an operator in a suitable Hilbert space? And how to spectral analyze T once the right Hilbert space H has been selected? Finally we characterize the stochastic processes that are governed by a single transfer operator.In our applications, the particular stochastic process will live on an infinite path-space which is realized in turn on a state space S. In the case of random walk on graphs G, S will be the set of vertices of G. The Hilbert space H on which the transfer operator T acts will then be an L
2 space on S, or a Hilbert space defined from an energy-quadratic form.This circle of problems is both interesting and non-trivial as it turns out that T may often be an unbounded linear operator in H; but even if it is bounded, it is a non-normal operator, so its spectral theory is not amenable to an analysis with the use of von Neumann’s spectral theorem. While we offer a number of applications, we believe that our spectral analysis will have intrinsic interest for the theory of operators in Hilbert space.

Keywords

EN

Hilbert space; spectrum; encoding; transfer operator; infinite graphs; wavelets; fractals; stochastic process; path measures; transition probability

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2010
• Volume: 8
• Issue: 3
• Pages: 340-363

Dates

published

1 - 6 - 2010

online

24 - 4 - 2010

Contributors

author

Palle Jorgensen

• Department of Mathematics, The University of Iowa, Iowa City, IA, 52242, USA

author

Myung-Sin Song

• Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL, 62026, USA

References

• [1] Z. Bai, J. Hou, J. Operat. Theor. 54, 291 (2005)
• [2] V. Baladi, Advanced Series in Nonlinear Dynamics, Vol. 16 (World Scientific, Singapore, 2000)
• [3] 0. Bratelli, P. Jorgensen, Wavelets Through a Looking Glass: The World of the Spectrum (Birkhäuser, Boston, 2002)
• [4] B. Brenken, P. Jorgensen, J. Operat. Theor. 25, 299 (1991)
• [5] D. Dutkay, P. Jorgensen, Rev. Mat. Iberoam. 22. 131 (2006)
• [6] H. Helson, The Spectral Theorem, vVol. 1227, Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1986)
• [7] T. Hida, Pitman Res. 310, 111 (1994)
• [8] T. Hida, Brownian Motion, Vol. 11, Appl. Math. (Springer-Verlag, New York, 1980)
• [9] T. Hida, NATO Adv. Sci. I. C-Mat. 449, 119 (1994)
• [10] D. Jakobson, I. Polterovich, Electron. Res. Announc. 11, 71 (2005) http://dx.doi.org/10.1090/S1079-6762-05-00149-6[Crossref]
• [11] P. Jorgensen, Graduate Texts in Mathematics, Vol. 234 (Springer, New York, 2006)
• [12] P. Jorgensen, E. Pearse, arXiv:0806.3881 [WoS]
• [13] P. Jorgensen, M.-S. Song, arXiv:0901.0195 [WoS]
• [14] P. Jorgensen, M.-S. Song, J. Math. Phys. 48, 103503 (2007) http://dx.doi.org/10.1063/1.2793569[Crossref]
• [15] A. Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer-Verlag, Berlin, 1977)
• [16] D. Labate, G. Weiss, E. Wilson, Contemp. Math. 345, 215 (2004)
• [17] M. Loève, Probability Theory. Foundations. Random sequences (D. Van Nostrand Company, Inc., Toronto-New York-London, 1955)
• [18] E. Nelson, J. Funct. Anal. 12, 211 (1973) http://dx.doi.org/10.1016/0022-1236(73)90025-6[Crossref]
• [19] M. Paluszynski, H. Šikic, G. Weiss, S. Xiao, Adv. Comput. Math. 18, 297 (2003) http://dx.doi.org/10.1023/A:1021312110549[Crossref]
• [20] I. Sadovnichaya, Differentsial’nye Uravneniya 42, 188 (2006)
• [21] M. Takeda, K. Tsuchida, T. Am. Math. Soc. 359, 4031 (2007) http://dx.doi.org/10.1090/S0002-9947-07-04149-9[Crossref]

Identifiers

DOI

10.2478/s11534-009-0119-4