Journal of Mathematical Physics
Feedback | Help | Exit
Search:
   
 
 
 
Previous Article
Integrable higher order deformations of Heisenberg supermagnetic model
The Heisenberg supermagnet model is an integrable supersymmetric system and has a close relationship with the strong electron correlated Hubbard model. In this paper, we investigate the integrable hig...
Next Article
On the Moyal deformation of Nahm equations in seven dimensions
We show how the reduced (anti-)self-dual Yang–Mills equations to seven dimensions described by the Nahm equations can be carried over to the Weyl–Wigner–Moyal formalism. In the proce...

Analysis of unbounded operators and random motion

J. Math. Phys. 50, 113503 (2009); doi:10.1063/1.3246837

Published 3 November 2009

You are not logged in to this journal. Log in

Palle E. T. Jorgensen
Department of Mathematics, The University of Iowa, 14 MLH, Iowa City, Iowa 52242-1419, USA
We study infinite weighted graphs with view to “limits at infinity” or boundaries at infinity. Examples of such weighted graphs arise in infinite (in practice, that means “very” large) networks of resistors or in statistical mechanics models for classical or quantum systems. However, more generally, our analysis includes reproducing kernel Hilbert spaces and associated operators on them. If X is some infinite set of vertices or nodes, in applications the essential ingredient going into the definition is a reproducing kernel Hilbert space; it measures the differences of functions on X evaluated on pairs of points in X. Moreover, the Hilbert norm-squared in [script H](X) will represent a suitable measure of energy. Associated unbounded operators will define a notion or dissipation, it can be a graph Laplacian or a more abstract unbounded Hermitian operator defined from the reproducing kernel Hilbert space under study. We prove that there are two closed subspaces in reproducing kernel Hilbert space [script H](X) that measure quantitative notions of limits at infinity in X: one generalizes finite-energy harmonic functions in [script H](X) and the other a deficiency index of a natural operator in [script H](X) associated directly with the diffusion. We establish these results in the abstract, and we offer examples and applications. Our results are related to, but different from, potential theoretic notions of “boundaries” in more standard random walk models. Comparisons are made. ©2009 American Institute of Physics
History: Received 7 May 2009; accepted 10 September 2009; published 3 November 2009
Permalink: http://link.aip.org/link/?JMAPAQ/50/113503/1
BUY THIS ARTICLE   (US$24)
Download PDF (282 kB) View Cart

KEYWORDS and PACS

Keywords
PACS

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
AIP is a member of CrossRef AIP

REFERENCES (35)
View in Separate Window/Tab
For access to fully linked references, you need to log in. For access to fully linked references, you need to Log in.
  1. Albeverio, S., Jorgensen, P. E. T., and Paolucci, A. M., “Multiresolution wavelet analysis of integer scale Bessel functions,” J. Math. Phys. 48, 073516 (2007).
  2. Alpay, D. and Levanony, D., “On the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions,” Potential Anal. 28, 163 (2008).
  3. Aplay, D. and Levanony, D., “Rational functions associated with the white noise space and related topics,” Potential Anal. 29, 195 (2008).
  4. Alpay, D., Shapiro, M., and Volok, D., “Reproducing kernel spaces of series of Fueter polynomials,” Operator Theory in Krein Spaces and Nonlinear Eigenvalue Problems, Operator Theory: Advances and Application Vol. 162 (Birkhäuser, Basel, 2006), pp. 19–45.
  5. Aronszajn, N., “Theory of reproducing kernels,” Trans. Am. Math. Soc. 68, 337(1950).
  6. Atkinson, K. and Han, W., Theoretical Numerical Analysis: A Functional Analysis Framework, Texts in Applied Mathematics Vol. 39, 2nd ed. (Springer, New York, 2005).
  7. Baladi, V., Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics Vol. 16 (World Scientific, River Edge, NJ, 2000).
  8. Barlow, M. T., Bass, R. F., Chen, Z. -Q., and Kassmann, M., “Non-local Dirichlet forms and symmetric jump processes,” Trans. Am. Math. Soc. 361, 1963 (2008).
  9. Brofferio, S. and Woess, W., “Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel-Leader graphs,” Ann. Inst. Henri Poincare 41, 1101 (2005).
  10. Choi, I. and Jorgensen, P., “C*-algebras generated by partial isometries,” Appl . Math. Comput. 5081, 1 (2008).
  11. Dunford, N. and Schwartz, J. T., Linear Operators. Part II: Spectral Theory, Self-Adjoint Operators in Hilbert space, Wiley Classics Library, with the assistance of W. G. Bade and R. G. Bartle(Wiley, New York, 1988) [reprint of the original Linear Operators. Part II, Wiley Classics Library (Wiley-Interscience, New York, 1963)].
  12. Dutkay, D. E., Han, D., Picioroaga, G., and Sun, Q., “Orthonormal dilations of Parseval wavelets,” Math. Ann. 341, 483 (2008).
  13. Dutkay, D. E. and Jorgensen,P. E. T., “Harmonic analysis and dynamics for affine iterated function systems,” Houst. J. Math. 33, 877 (2007).
  14. Dutkay, D. E. and Jorgensen, P. E. T., “Martingales, endomorphisms, and covariant systems of operators in Hilbert space,” J. Oper. Theory 58, 269 (2007).
  15. Dutkay, D. E. and Jorgensen, P. E. T., “A duality approach to representations of Baumslag-Solitar groups,” Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey, Contemporary Mathematics Vol. 449 (American Mathematics Society, Providence, RI, 2008), pp. 99–127.
  16. Radon Transforms, Geometry, and Wavelets, Contemporary Mathematics Vol. 464, edited by G. Ólafsson, E. L. Grinberg, D. Larson, P. Jorgensen, P. R. Massopust, E. Quinto, and B. Rubin (American Mathematics Society, Providence, RI, 2008), pp. 75–101.
  17. Dutkay, D. E. and Jorgensen, P. E. T., “Quasiperiodic spectra and orthogonality for iterated function system measures,” Math. Z. 261, 373 (2009).
  18. Hida, T., Brownian Motion, Applications of Mathematics Vol. 11 (Springer-Verlag, New York, 1980), translated from the Japanese by the author and T. P. Speed.
  19. Jorgensen, P. E. T., Kornelson, K., and Shuman, K., “Harmonic analysis of iterated function systems with overlap,” J. Math. Phys. 48, 083511 (2007),.
  20. Jorgensen, P. E. T., “Orthogonal exponentials for Bernoulli iterated function systems,” Representations, Wavelets, and Frames, Applied and Numerical Harmonic Analysis (Birkhäuser, Boston, MA, 2008), pp. 217–237.
  21. Jorgensen, P. E. T. and Mohari, A., “Localized bases in L2(0,1) and their use in the analysis of Brownian motion,” J. Approx. Theory 151, 20 (2008).
  22. Jorgensen, P. E. T. and Pearse, E. P. J., “Operator theory of electrical resistance networks,” e-print arXiv:0806.3881v1 (math.OA).
  23. Jorgensen, P. E. T. and Song, M. -S. “Entropy encoding, Hilbert space, and Karhunen-Loève transforms,” J. Math. Phys. 48, 103503 (2007).
  24. Kolmogoroff, A., Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer-Verlag, Berlin, 1977) [reprint of Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer-Verlag, Berlin, 1933)].
  25. Lax, P. D. and Phillips, R. S. “Scattering theory for automorphic functions,” Bull., New Ser., Am. Math. Soc. 2, 261 (1980).
  26. Nelson, E., “The free Markoff field,” J. Funct. Anal. 12, 211 (1973).
  27. Ortner, R. and Woess, W., “Non-backtracking random walks and cogrowth of graphs,” Can. J. Math. 59, 828 (2007).
  28. Reed, M. and Simon, B., Fourier Analysis, Self-Adjointness, Methods of Modern Mathematical Physics Vol. 2 (Academic, New York/Harcourt Brace Jovanovich, New York, 1975).
  29. Song, M. -S., “Wavelet image compression,” Operator Theory, Operator Algebras, and Applications, Contemporary Mathematics Vol. 414 (American Mathematics Society, Providence, RI, 2006), pp. 41–73.
  30. P. E. T. Jorgensen, K. Kornelson, and K. Shuman, “Entropy encoding in wavelet image compression,” Representations, Wavelets, and Frames, Applied and Numerical Harmonic Analysis (Birkhäuser, Boston, MA, 2008), pp. 293–311.
  31. Stone, M. H., Linear Transformations in Hilbert Space, American Mathematical Society Colloquium Publications Vol. 15 (American Mathematical Society, Providence, RI, 1990) [reprint of Linear Transformations in Hilbert Space, American Mathematical Society Colloquium Publications (American Mathematical Society, Providence, RI, 1932)].
  32. Stone, H. M., von Neumann, J., Friedrichs, K., and Kato, T. (unpublished).
  33. von Neumann, J., “Über adjungierte Funktionaloperatoren,” Ann. Math. 33, 294 (1932).
  34. Yamasaki, K. and Nagahama, H., “Energy integral in fracture mechanics (J-integral) and Gauss-Bonnet theorem,” Z. Angew. Math. Mech. 88, 515 (2008).
  35. Zhang, H., “Orthogonality from disjoint support in reproducing kernel Hilbert spaces,” J. Math. Anal. Appl. 349, 201 (2009).

CITING ARTICLES
For access to citing articles, you need to log in.
For access to citing articles, you need to Log in.


Copyright © American Institute of Physics