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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

On stochastic Hamiltonian mechanics for diffusion processes

Tetsuya Misawa

Published Jan 1, 1986• 6 citations
Publisher
DOI
10.1007/bf02722218

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It is shown how Adler's trace dynamics can be applied to stochastic mechanics and other complex classical dynamical systems. Emergent non-commutivity due to the fractal nature of sample trajectories is closely related to the fact that the forward and backward time derivatives are different for these diffusions. A new variational approach to stochastic mechanics based on trace dynamics is introduced. It is shown that Yasue's method and Guerra and Morato's method can both be generalized to allow for any diffusion constant in a stochastic model of Schrodinger's equation, and that they can all also describe dissipative diffusion. Then it is shown that the trace dynamical theory seems to only describe dissipative diffusion unless an extra quantum mechanical potential term is added to the Hamiltonian. The differential space theory of Wiener and Siegel is reconsidered as a useful tool in this framework, and is generalized to stochastic processes instead of deterministic ones for the hidden trajectories of observables. It is proposed that the natural measure space for Wiener-Siegel theory is Haar measure for random unitary matrices. A new interpretation of the polychotomic algorithm is given.
A diffusion process of which the diffusion property is preserved under time reversal is called a time-reversible diffusion process. A typical example of such a process is found in Nelson’s stochastic mechanics. Within the extended framework of stochastic integrals proposed by Itô, a sort of the interpolation formula, called a stochastic dynamical interpolation formula, to the time-reversible diffusion process, is derived from a pinned process for the diffusion process. It is shown that the stochastic interpolation formula is regarded as an extention of Lévy’s interpolation formula, which is obtained on the basis of a pinned Brownian motion.