Classical stochastic processes from quantum stochastic calculus
Using multidimensional quantum stochastic calculus, the author constructs a (weak) one-parameter representation \(\tilde{j}_t\), \(t\in {\mathbb R}_+\), of the Lie algebra \(gl(N)\) of \(N\times N\) matrices, which is then extended to a representation \(\tilde{J}_t\), \(t\in {\mathbb R}_+\), of the universal enveloping algebra \({\mathcal U}(gl(N))\) of \(gl(N)\). Given a self-adjoint Casimir element \(Z\) in the center \({\mathcal Z} ({\mathcal U}(gl(N)))\) of \({\mathcal U}(gl(N))\), a classical stochastic process in the vacuum state is constructed as \(Z_t = \tilde{J}_tZ\), \(t\in {\mathbb R}_+\). When \(Z\) is a symmetric polynomial one obtains new integer-valued classical processes, and the classical Poisson process as a particular case. Chaotic expansions in multiple quantum stochastic integrals are also obtained for these processes.
scientific article; zbMATH DE number 1756305