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Birkhäuser Basel eBooks

The Itô Calculus

Masao Nagasawa

Published Jan 1, 2000• 0 citations
Publisher

Abstract

We need Itô’s stochastic calculus in the theory of diffusion processes for non-relativistic quantum particles. The theory of martingales, which is the right language for the stochastic calculus, is briefly explained in Section 16.2. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-0348-8383-2_16

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We need Itô’s stochastic calculus in the theory of diffusion processes for non-relativistic quantum particles. The theory of martingales, which is the right language for the stochastic calculus, is briefly explained in Section 16.2. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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The equation of motion of a quantum particle(s) in a potential field will be introduced. The equation of motion should determine drift fields a (t, x) and the movement X t of quantum particles, accordingly. However, the interrelation between the equation of motion and the movement X t of a particle(s) is not so direct as in classical mechanics, and we must go into deeper mathematical structures of diffusion processes to clarify the relation. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Bernstein diffusions belong to a new class of time symmetric (but not time homogeneous) stochastic processes associated with the quantum dynamics of nonrelativistic particles in potentials. It is shown in which sense Feynman’s formal path integral method can be interpreted in terms of those processes, specially for the subset of Gaussian Bernstein diffusions. The familiar Ornstein-Uhlenbeck process becomes, in this framework, a particular Gaussian process in a large class of Bernstein diffusions, all associated with the same dynamics. The method is also illustrated for some Hilbert space valued Gaussian Diffusions These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. The Schroedinger equation and the so-called Nelson equations are used to determine the diffusion tensor and the drift vectors characteristic of such a process. It is then possible to write down the forward and backward Kolmogorov equations that are used to determine the conditional probability density, as well as the Fokker-Planck equation that is used to determine the normal probability density. This method is applied to several simple cases and the results obtained are compared with those of the orthodox theory. Among the most important differences let us mention that any system evolving freely from any original state returns spontaneously to its ground state, that the definitions of the particles velocities and momenta are impossible because the sample functions of a diffusion Markov process nowhere possess a derivative with respect to time and that the so-called collapse of the wave function is a mere mirage explained by the updating choice between new and older initial conditions in the resolution of partial differential equations. We finally attempt to extend the proposed theory to the domain of the relativistic quantum mechanics. It is promising but is clearly unfinished because it has not been possible up to now to solve the Nelson and Kolmogorov equations, except in the very simple case of a free particle.