On the Kinematics of Stochastic Mechanics
In a series of recent papers [P1]-[P3] we have shown that the complexification of the velocity and momentum processes permits to effectively develop a Lagrangian and Hamiltonian formalism in stochastic mechanics [N1], [G], [N2]. In this paper, the kinematics employed in [P1]-[P3] is more thoroughly analyzed, particularly from the probabilistic viewpoint. The outline of the paper is as follows. In Section 2, we introduce the appropriate kinematics for finite-energy diffusions developing on [N3]. In Section 3, we discuss in detail the properties of the quantum noise and derive a fundamental change of variables formula. In the following section, we consider the Markov case. In Section 5, for the purpose of comparison, we give a strong form of the classical Hamilton principle. In Section 6, we present the quantum Hamilton principle.
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