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We introduce the concepts of Poisson brackets for classical noise, and ofcanonically conjugate Wiener processes (symplectic noise). Phase spacediffusions driven by these processes are considered and the general form of astochastic process preserving the full (system and noise) Poisson structure isobtained. We show that, once the classical stochastic model is required topreserve the joint system and noise Poisson bracket, it has much in common withquantum markovian models.

Symplectic Noise & The Classical Analog of the Lindblad Generator