We formulate the quantum system of an oscillator driven by a quantum Wiener process, in the locally convex setting based on the rigged triple S(ℝ) ⊂ _L_2(ℝ) ⊂ S'(ℝ). The generalized observables are taken to be the elements of ℒ(S(ℝ), S'(ℝ)). Pulling the dynamics back to phase space by means of the inverse of Weyl quantization, we prove that the time translations semigroup is equicontinuous of class _C_0. Moreover, it is differentiable, and its generator is an extension to ℒ(S(ℝ), S'(ℝ)) of the known result for bounded operators.
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Daniel A. Dubin, Mark A. Hennings, The Damped Oscillator: A Locally Convex Formulation. Publ. Res. Inst. Math. Sci. 41 (2005), no. 1, pp. 11–36DOI 10.2977/PRIMS/1145475403