Quantum independent increment processes on superalgebras
Starting from the concept of quantum stochastic processes in the sense of the first author, \textit{A. Frigerio} and \textit{J. T. Lewis} [Publ. Res. Inst. Math. Sci. 18, 97-133 (1982; Zbl 0498.60099)], the notion of quantum independent increment processes on superalgebras is introduced. A reconstruction theorem is proved establishing a one-to-one correspondence between these processes and their infinitesimal generators, which are conditionally positive linear functionals on an involutive bialgebra. (Bialgebras sometimes are called Hopf-algebras.) In particular a new technique for constructing continuous tensor products of \(Z_ 2\)-graded involutive algebras not based on the representations of the canonical (anti-)commutation relations is provided. A quantum version of the Lévy-Khintchine formula and a full classification of the ``continuous trajectories'' quantum processes with independent stationary increments are obtained. It is proved that solutions of quantum stochastic differential equations on the Fock space over \(L_ 2(R_ +)\) in the sense of \textit{R. L. Hudson} and \textit{K. R. Parthasarathy} [Commun. Math. Phys. 93, 301-323 (1984; Zbl 0546.60058)], give rise to quantum independent increment processes as defined in the paper. A formula for the generators of these special processes is derived.
quantum independent increment processes on superalgebras
quantum stochastic differential equations
scientific article; zbMATH DE number 4020082
scientific article