Some properties of quantum Lévy area in Fock and non-Fock quantum stochastic calculus

• Autorzy: Chen S. , Hudson R.
• Języki publikacji: EN

Abstrakty

EN

We consider the analogue of Lévy area, defined as an iterated stochastic integral, obtained by replacing two independent component onedimensional Brownian motions by the mutually non-commuting momentum and position Brownian motions P and Q of either Fock or non-Fock quantum stochastic calculus, which are also stochastically independent in a certain sense. We show that the resulting quantum Lévy area is trivially distributed in the Fock case, but has a non-trivial distribution in non-Fock quantum stochastic calculus which, after rescaling, interpolates between the trivial distribution and that of classical Lévy area in the “infinite temperature” limit. We also show that it behaves differently from the classical Lévy area under a kind of time reversal, in both the Fock and non-Fock cases.

Słowa kluczowe

EN

quantum Lévy area; non-Fock quantum stochastic calculus; time reversal

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2013
• Tom: Vol. 33, Fasc. 2
• Strony: 425--434
• Opis fizyczny: Bibliogr. 10 poz., tab.

Twórcy

autor

Chen S.

• Department of Computer Science, Loughborough University, Loughborough, LE11 3TU, Great Britain
• Department of Mathematics, Loughborough University, Loughborough, LE11 3TU, Great Britain

autor

Hudson R.

• Department of Mathematics, Loughborough University, Loughborough, LE11 3TU, Great Britain

Bibliografia

• [1] A. M. Cockcroft and R. L. Hudson, Quantum mechanical Wiener processes, J. Multivariate Anal. 7 (1977), pp. 107-124.
• [2] P. B. Cohen, T. W. M. Eyre, and R. L. Hudson, Higher order Itô product formula and generators of evolutions and flows, Internat. J. Theoret. Phys. 34 (1995), pp. 1-6.
• [3] R. L. Hudson, Sticky shuffle Hopf algebras and their stochastic representations, in: New Trends in Stochastic Analysis and Related Topics. A Volume in Honour of K. D. Elworthy, H. Zhao and A. Truman (Eds.), World Scientific, 2012, pp. 165-181.
• [4] R. L. Hudson, Quantum Lévy area as a quantum martingale limit, in: Quantum Probability and Related Topics: Proceedings of the 32nd Conference, Levico Terme, Italy, 29 May-4 June 2011, L. Accardi and F. Fagnola (Eds.), World Scientific, 2013, pp. 169-188.
• [5] R. L. Hudson and J. M. Lindsay, A non-commutative martingale representation theorem for non-Fock quantum Brownian motion, J. Funct. Anal. 61 (1985), pp. 202-221.
• [6] R. L. Hudson and K. R. Parthasarathy, Quantum Itô’s formula and stochastic evolutions, Comm. Math. Phys. 93 (1984), pp. 301-323.
• [7] N. Ikeda and S. Taniguchi, The Itô-Nisio theorem, quadratic Wiener functionals and 1-solitons, Stochastic Process. Appl. 120 (2010), pp. 605-621.
• [8] P. Lévy, Wiener’s random function and other Laplacian random functions, in: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. Berkeley 1950, University of California Press, 1951, pp. 171-187.
• [9] J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. London Math. Soc. 9 (1934), pp. 6-13.
• [10] K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel 1992.

Uwagi

Dedicated to the memory of Slava Belavkin

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).