Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 5 (2000) > Paper 17 open journal systems 


Mild Solutions of Quantum Stochastic Differential Equations

Franco Fagnola, Università di Genova
Stephen J. Wills, University of Nottingham


Abstract
We introduce the concept of a mild solution for the right Hudson-Parthasarathy quantum stochastic differential equation, prove existence and uniqueness results, and show the correspondence between our definition and similar ideas in the theory of classical stochastic differential equations. The conditions that a process must satisfy in order for it to be a mild solution are shown to be strictly weaker than those for it to be a strong solution by exhibiting a class of coefficient matrices for which a mild unitary solution can be found, but for which no strong solution exists.


Full text: PDF

Pages: 158-171

Published on: September 12, 2000
Bibliography
  1. S. Attal and J.M. Lindsay, Quantum stochastic calculus with maximal operator domains, Preprint 1997 (to appear in Ann Probab).
  2. B.V. Rajarama Bhat and K.B. Sinha, Examples of unbounded generators leading to nonconservative minimal semigroups, in, Quantum Probability and Related Topics IX, ed. L Accardi, World Scientific, Singapore (1994), 89-103.
  3. A.M. Chebotarev and F. Fagnola, Sufficient conditions for conservativity of minimal quantum dynamical semigroups, J Funct Anal 153 (1998), 382-404. Math. Review 99d:81064
  4. G. Da Prato, M. Iannelli and L. Tubaro, Some results on linear stochastic differential equations in Hilbert spaces Stochastics6 (1981/82), 105-116. Math. Review 83m:60076
  5. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, Cambridge, (1992). Math. Review 95g:60073
  6. E.B. Davies, One-parameter Semigroups, Academic Press, London (1980). Math. Review 82i:47060
  7. F. Fagnola, Characterization of isometric and unitary weakly differentiable cocycles in Fock space, in, Quantum Probability and Related Topics VIII, ed. L Accardi, World Scientific, Singapore (1993), 143-164. Math. Review 95e:81122
  8. F. Fagnola, Quantum Markov Semigroups and Quantum Flows, PhD thesis, Scuola Normale Superiore di Pisa (1999); published in Proyecciones 18 (1999), 1-144.
  9. F. Fagnola and S.J. Wills, Solving quantum stochastic differential equations with unbounded coefficients, Preprint 1999.
  10. R.L. Hudson and K.R. Parthasarathy, Quantum Itô's formula and stochastic evolutions, Comm Math Phys 93 (1984), 301-323. Math. Review 86e:46057
  11. P-A Meyer, Quantum Probability for Probabilists, 2nd Edition, Springer Lecture Notes in Mathematics 1538 Heidelberg (1993). Math. Review 94k:81152
  12. A. Mohari, Quantum stochastic differential equations with unbounded coefficients and dilations of Feller's minimal solution, Sankhya Ser A 53 (1991), 255-287. Math. Review 93m:81083
  13. K.R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel (1992). Math. Review 93g:81062













Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X