Uncertainty principle for generalized diffusions
- 42 Downloads
It is shown that with any generalized diffusion a complementary variable, theprobabilistic group velocity (PGV), can be associated, such that the uncertainties in the position (diffusion) and its complementary PGV (like momentum) satisfy a Heisenberg-type uncertainty relation. It is shown that the bound is achieved in the linear Gaussian case. In the statistical steady state, the PGV vanishes identically. The uncertainty in the PGV is an achievable upper bound to the rate of the RMS value of the diffusion. The PGV is further related to the entropy rate of the diffusion process.
KeywordsEntropy Steady State Field Theory Elementary Particle Quantum Field Theory
Unable to display preview. Download preview PDF.
- Blanchard, P., Combe, P., and Zheng, W. (1987).Mathematical and Physical Aspects of Stochastic Mechanics, Springer-Verlag, Berlin.Google Scholar
- Carlen, E. A. (1988). Progress and problems in stochastic mechanics, inStochastic Methods in Mathematics and Physics, R. Gielerak and W. Karwowski, eds., World Scientific, Singapore.Google Scholar
- Costa, M. H. M., and Cover, T. M. (1984). On the similarity of the entropy power inequality and the Brunn-Minkowski inequality,IEEE Transactions on Information Theory,IT-30(6).Google Scholar
- Fürth, R. (1933).Zeitschrift für Physik,81, 143–162.Google Scholar
- Hale, J. K., Magalhaes, L. T., Oliva, W. M., and Rybakowski, K. (1984).An Introduction to Infinite Dimensional Dynamical Systems, Springer-Verlag, Vienna.Google Scholar
- Verriest, E. I., and Shin, D.-R. (1991). An uncertainty principle for continuous semimartingales, inProceedings of the IEEE International Symposium on Information Theory, Budapest, Hungary, June 1991, p. 170.Google Scholar