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Construction of quantum diffusions. (English) Zbl 0542.60053

Quantum probability and applications to the quantum theory of irreversible processes, Proc. int. Workshop, Villa Mondragone/Italy 1982, Lect. Notes Math. 1055, 173-198 (1984).
[For the entire collection see Zbl 0527.00022.]
This work reviews in part non-rigorously the theory of stochastic evolutions and Boson quantum stochastic differential equations as it stood in 1982. Quantum Brownian motion \((A_ t,t\geq 0)\) is defined abstractly by Boson commutation rules and expectation values; the standard quantum Brownian motion is that given in Fock space by \(A_ t=a(\chi_{[0,t]})\). Stochastic integrals of the form \(M(t)=\int^{t}_{0}(dA^{†}F+GdA+Hd\tau)\) are defined by the formula for matrix elements between exponential vectors \[ <\psi_ f,M(t)\psi_ g>=\int^{t}_{0}<\psi_ f,(\bar f(\tau)F(\tau)+g(\tau)G(\tau)+H(\tau))\psi_ g>d\tau. \] The Weyl operators \(W_ t(f)=W(f\chi_{[0,t]})\) are then the solution of the stochastic differential equation \[ dW_ t(f)=W_ t(f)(dA^{†}f- \bar fdA-{1\over2}| f|^ 2dt),\quad W_ 0(f)=I; \] at the same time they are stochastic product integrals \[ W_ t(f)=\exp(\int^{t}_{0}(dA^{†}f- fdA))=\prod^{t}_{0}\exp(dA^{†}f-fdA). \] This suggests the construction of unitary processes solving \(dU=U(dA^{†}L- L^{†}dA+(iH-{1\over2}L^{†}L)dt)\), \(U(0)=I\), as stochastic product integrals \(\prod^{t}_{0}\exp(dA^{†}L-L^{†}dA+iH dt)\), the latter being constructed heuristically by a discrete approximation. The Ito product formula is then a generalisation of the Weyl relation. Quantum diffusions are solutions of stochastic differential equations of the form \(da=FdA^{†}+GdA+Hdt\) for an operator a required to satisfy the Boson commutation relation \([a,a^{†}]=I\); formally they are shown to be of the form \(a_ t=U_ ta_ 0U_ t^{-1}\) where U is a unitary process. A completely positive semigroup \((J_ t,t\geq 0)\) and a contraction semigroup \((T_ t,t\geq 0)\) are generated by the formulae \(J_ t(X)={\mathbb{E}}_ 0(U_ tXU_ t^{-1})\), \(T_ t={\mathbb{E}}_ 0(U_ t)\) where \({\mathbb{E}}\) is the vacuum conditional expectation, and give noncommutative generalisations of the solution of elliptic Cauchy problems by stochastic differential equations and of the Feynman-Kac formula, respectively. Quantum diffusions in which the coefficients F,G,H are inhomogeneous linear polynomials in a and \(a^{†}\) are shown to reduce to six canonical forms which are solved explicitly.

MSC:

60Hxx Stochastic analysis
60J60 Diffusion processes
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
81P20 Stochastic mechanics (including stochastic electrodynamics)

Citations:

Zbl 0527.00022