Summary
We show that a gradient operator defined by perturbations of the Poisson process jump times can be used with its adjoint operator instead of the annihilation and creation operators on the Poisson-Charlier chaotic decomposition to represent the Poisson process. The quantum stochastic integration and the Itô formula are developed accordingly, leading to commutation relations which are different from the CCR. An analog of the Weyl representation is defined for a subgroup ofSL(2, ℝ), showing that the exponential and geometric distributions are closely related in this approach.
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Privault, N. A different quantum stochastic calculus for the Poisson process. Probab. Th. Rel. Fields 105, 255–278 (1996). https://doi.org/10.1007/BF01203837
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DOI: https://doi.org/10.1007/BF01203837
Mathematics Subject Classification (1991)
- 81S25
- 60H07
- 60G55