×

Sesquilinear quantum stochastic analysis in Banach space. (English) Zbl 1308.81125

Summary: A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie-Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.

MSC:

81S25 Quantum stochastic calculus
46L60 Applications of selfadjoint operator algebras to physics
22E70 Applications of Lie groups to the sciences; explicit representations
35R60 PDEs with randomness, stochastic partial differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Biane, Ph., Calcul stochastique non-commutatif, (Lectures on Probability Theory. Lectures on Probability Theory, Lecture Notes in Math., vol. 1608 (1995), Springer-Verlag: Springer-Verlag Berlin), 1-96 · Zbl 0878.60041
[2] Dales, H. G., (Banach Algebras and Automatic Continuity. Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, vol. 24 (2001), Oxford University Press: Oxford University Press Oxford) · Zbl 0981.46043
[5] Evans, M. P., Existence of quantum diffusions, Probab. Theory Related Fields, 81, 4, 473-483 (1989) · Zbl 0667.60060
[6] Gough, J.; James, M., The series product and its application to quantum feedforward and feedback networks, IEEE Trans. Automat. Control, 54, 11, 2530-2544 (2009) · Zbl 1367.81026
[7] Guichardet, A., (Symmetric Hilbert Spaces and Related Topics. Infinitely Divisible Positive Definite Functions. Continuous Products and Tensor Products. Gaussian and Poissonian Stochastic Processes. Symmetric Hilbert Spaces and Related Topics. Infinitely Divisible Positive Definite Functions. Continuous Products and Tensor Products. Gaussian and Poissonian Stochastic Processes, Lecture Notes in Math., vol. 261 (1972), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0265.43008
[8] Hudson, R. L.; Parthasarathy, K. R., Quantum Itô’s formula and stochastic evolution, Comm. Math. Phys., 93, 3, 301-323 (1984) · Zbl 0546.60058
[9] Lindsay, J. M., Quantum stochastic analysis—an introduction, (Franz, U.; Schürmann, M., Quantum Independent Increment Process, I: From Classical Probability to Quantum Stochastic Calculus. Quantum Independent Increment Process, I: From Classical Probability to Quantum Stochastic Calculus, Lecture Notes in Math., vol. 1865 (2005), Springer-Verlag: Springer-Verlag Heidelberg), 181-271 · Zbl 1072.81039
[11] Lindsay, J. M.; Parthasarathy, K. R., On the generators of quantum stochastic flows, J. Funct. Anal., 158, 2, 521-549 (1998) · Zbl 0914.60033
[12] Lindsay, J. M.; Sinha, K. B., A quantum stochastic Lie-Trotter product formula, Indian J. Pure Appl. Math., 41, 1, 313-325 (2010) · Zbl 1196.81153
[13] Lindsay, J. M.; Skalski, A. G., Quantum stochastic convolution cocycles II, Comm. Math. Phys., 280, 3, 575-610 (2008) · Zbl 1149.81015
[14] Lindsay, J. M.; Skalski, A. G., On quantum stochastic differential equations, J. Math. Anal. Appl., 330, 1093-1114 (2007) · Zbl 1175.81140
[15] Lindsay, J. M.; Wills, S. J., Existence of Feller cocycles on a \(C^\ast \)-algebra, Bull. London Math. Soc., 33, 5, 613-621 (2001) · Zbl 1032.46088
[16] Meyer, P.-A., (Quantum Probability for Probabilists. Quantum Probability for Probabilists, Lecture Notes in Math., vol. 1538 (1995), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0877.60079
[17] Parthasarathy, K. R., (An Introduction to Quantum Stochastic Calculus. An Introduction to Quantum Stochastic Calculus, Monographs in Mathematics, vol. 85 (1992), Birkhäuser: Birkhäuser Basel) · Zbl 0751.60046
[18] Pisier, G., Exact operator spaces, (Recent Advances in Operator Algebras (Orléans, 1992). Recent Advances in Operator Algebras (Orléans, 1992), Astérisque, vol. 232 (1995)), 159-186 · Zbl 0844.46031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.