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arXiv:0804.1948

Characterization of unitary processes with independent and stationary increments

Lingaraj Sahu, Kalyan B. Sinha

Published Apr 11, 2008• 1 citation
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Abstract

This is a continuation of the earlier work \cite{SSS} to characterize stationary unitary increment Gaussian processes. The earlier assumption of uniform continuity is replaced by weak continuity and with a technical assumption on the domain of the generator, unitary equivalence of the processes to the solution of Hudson-Parthasarathy equation is proved.

DOI
10.48550/arxiv.0804.1948

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This is a continuation of the earlier work \cite{SSS} to characterize stationary unitary increment Gaussian processes. The earlier assumption of uniform continuity is replaced by weak continuity and with a technical assumption on the domain of the generator, unitary equivalence of the processes to the solution of Hudson-Parthasarathy equation is proved.
Many examples of quantum independent stationary increment processes are solutions of quantum stochastic differential equations. We give a common characterization of these examples by a quantum stochastic differential equation on an abstract *-bialgebra. Specializing this abstract *-bialgebra and the coefficients of the equation, we obtain the equations for the Unitary Noncommutative Stochastic processes of [12], the Quantum Wiener Process [2], the Azéma martingales [11] and for other examples. The existence and uniqueness of a solution of the general equation is shown. Assuming the boundedness of this solution, we prove that it is a continuous and stationary independent increment process.
In the calculus of Hudson and Parthasarathy we show the existence of a solution to the quantum stochastic evolution equation U ( t ) = I + ∫ 0 t U ( s ) ( L 1 d Λ ( s ) + L 2 d A ( s ) + L 3 d A † ( s ) + L 4 d s ) for a class of unbounded operators L1, …, L4 in the initial space, and show that the necessary and sufficient condition that U be a unitary process is the same as in the case that L1, …, L4 are bounded. Applications of the vacuum conditional expectation to U give rise to strongly continuous semigroups and their generators in the initial space, and to ultraweakly continuous completely positive semigroups and their generators in the von Neumann algebra of the initial space.