On q-deformed quantum stochastic calculus

• Autorzy: Sniady P.
• Języki publikacji: EN

Abstrakty

EN

In this paper we investigate a quantum stochastic calculus built of creation, annihilation and number of particles operators which fulfill some deformed com-mutation relations. Namely, we introduce a deformation of a number of particles operator which hassimple commutation relations with well-known q-deformed creation and annihilation operators. Since all operators considered in this theory are bounded, we do not dealwith some difficulties of a non-deformed theory of Hudson and Parthasarathy [8]. We define stochastic integrals and estimate them in the operator norm. We prove Itô’s formula as well.

Słowa kluczowe

EN

stochastic calculus; particles operators; theory of Hudson and Parthasarathy

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2001
• Tom: Vol. 21, Fasc. 1
• Strony: 231--251
• Opis fizyczny: Bibliogr.10 poz.

Twórcy

autor

Sniady P.

• Institute of Mathematics, WrocIaw Universib pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Bibliografia

• [1] B. Barnett, R. F. Streater and L F. Wilde, The Itô-Cliffbrd integral, J. Funct. Anal. 48 (1982), pp. 172-212.
• [2] B. Barnett, R. F. Streater and I. F. Wilde, Quasi-free quantum stochastic integrals for the CAR and CCR, J. Funct. Anal. 52 (1983), pp. 19-47.
• [3] P. Biane and R. Speicher, Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, preprint, 1997, Probab, Theory Related Fields (to appear).
• [4] M. Bożejko, Completely positive maps on Coxeter groups and the ultracontractivity of the q-Ornstein-Uhlenbeck semigroup, Banach Center Publ. 43 (1998), pp. 87-93.
• [5] M. Bożejko and R. Speicher, An example of a generalized Brownian motion. 11, in: Quantum Probability and Related Topics VII, L. Accardi (Ed.), World Scientific, Singapore 1992, pp. 219-236.
• [6] M. Bożejko and R. Speicher, Completely positive maps on Coxeter, deformed commutation relations, and operator spaces, Math. Ann. 300 (1994), pp. 97-120.
• [7] U. Frisch and R. Bourret, Parastochastics, J. Math. Phys. 11 (1970), pp. 364-390.
• [8] R. L. Hudson and K. R. Parthasarathy, Quantum Itô's formula and stochastic evolution, Comm. Math. Phys. 93 (1994), pp. 301-323.
• [9] B. Kümmerer and R. Speicher, Stochastic integration on the Cuntz algebra Oœ, J. Funct. Anal. 103 (1992), pp. 372-408.
• [10] R. Speicher, Stochastic Integration on the Full Fock Space with the Help of a Kernel Calculus, Publ. Res. Inst. Math. Sci., Kyoto University, 27, No. 1 (1991).