Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Quantum Ω-semimartingales and stochastic evolutions
AU - Belton, Alexander C. R.
PY - 2001/12/1
Y1 - 2001/12/1
N2 - We explore Ω-adaptedness, a variant of the usual notion of adaptedness found in stochastic calculus. It is shown that the (non-adapted) quantum stochastic integrals of bounded, Ω-adapted processes are themselves bounded and Ω-adapted, a fact that may be deduced from the Bismut–Clark–Ocone formula of Malliavin calculus. An algebra analogous to Attal's class of regular quantum semimartingales is defined, and product and functional Itô formulae are given. We consider quantum stochastic differential equations with bounded, Ω-adapted coefficients that are time dependent and act on the whole Fock space. Solutions to such equations may be used to dilate quantum dynamical semigroups in a manner that generalises, and gives new insight into, that of R. Alicki and M. Fannes (1987, Comm. Math. Phys.108, 353–361); their unitarity condition is seen to be the usual condition of R. L. Hudson and K. R. Parthasarathy (1984, Comm. Math. Phys93, 301–323).
AB - We explore Ω-adaptedness, a variant of the usual notion of adaptedness found in stochastic calculus. It is shown that the (non-adapted) quantum stochastic integrals of bounded, Ω-adapted processes are themselves bounded and Ω-adapted, a fact that may be deduced from the Bismut–Clark–Ocone formula of Malliavin calculus. An algebra analogous to Attal's class of regular quantum semimartingales is defined, and product and functional Itô formulae are given. We consider quantum stochastic differential equations with bounded, Ω-adapted coefficients that are time dependent and act on the whole Fock space. Solutions to such equations may be used to dilate quantum dynamical semigroups in a manner that generalises, and gives new insight into, that of R. Alicki and M. Fannes (1987, Comm. Math. Phys.108, 353–361); their unitarity condition is seen to be the usual condition of R. L. Hudson and K. R. Parthasarathy (1984, Comm. Math. Phys93, 301–323).
KW - Ω-adaptedness
KW - quantum semimartingales
KW - quantum stochastic differential equations
KW - quantum dynamical semigroups
U2 - 10.1006/jfan.2001.3809
DO - 10.1006/jfan.2001.3809
M3 - Journal article
VL - 187
SP - 94
EP - 109
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 1
ER -