Elements of the Theory of Stochastic Processes

J Procházka

Abstract Both the corpuscular theory of light and the theory of stochastic processes are well known in the literature. However, they are not systematically used together for description of optical phenomena. There are optical phenomena, such as the well-known three-polarizer experiment or other phenomena related to polarization of light, which have never been quantitatively and qualitatively explained using the concept of quantum of light (photon). The situation changed in 2022 when stochastic memoryless and independent (IM) process formulated within the framework of the theory of stochastic processes was introduced. It is suitable for determination of probability (density) functions characterizing interaction (transmission or reflection) of individual photons with optical elements on the basis of experimental data. The process has memoryless (Markov) property, and it is assumed that the interactions of individual photons with an optical system are independent. Formulae needed for analysis of data in the context of polarization of light are derived. An example analysis of the three-polarizer experiment is performed and numerical result of the probability (density) functions are determined. These original results were missing in the literature. The results imply that the possibilities of the corpuscular theory of light to describe optical phenomena can be significantly extended with the help of stochastic IM process and the theory of stochastic processes in general.

A. O. Bolivar

Aubrey Truman, Ian M Davies

Luis de la Peña, Ana Marı́a Cetto, Andrea Valdés-Hernández

The connection is established between two theories that have developed independently with the aim to describe quantum mechanics as a stochastic process, namely stochastic quantum mechanics (sqm) and stochastic electrodynamics (sed). Important commonalities and complementarities between the two theories are identified, notwithstanding their dissimilar origins and approaches. Further, the dynamical equation of sqm is completed with the radiation terms that are an integral element in sed. The central problem of the transition to the quantum dynamics is addressed, pointing to the key role of diffusion in the emergence of quantization.

Yuri A. Rylov

Edward W. Barankin

This author has been engaged over the past several years in the effort to develop certain basic intuitions into a fully mathematical theory. These evolving intuitions first began with the idea that the economist's notion of utility might more truly be represented mathematically by a normed measure on the commodity space rather than by a point function, the value of the measure on a particular set of goods having the sense of a relative propensity of the individual to select a good in that set. With this idea there came in turn, quite naturally, the insight that utility and probability are one and the same thing. And accordingly it emerged that the theory we were seeking would not be restricted to the economic behavior of human beings, nor even to the behavior in general of human beings: it would pertain, in utmost generality, to all behavior of all systems. Thus, it must affirm that all behavior is the evolution of stochastic processes, and that all systems are stochastic processes. Consistent with this the structural elements of stochastic processes are seen as the fundamental, objective, real entities, the ultimate building-blocks of reality. These are eventualities and acts and-according to the traditional mathematical representation of stochastic processes, also-probabilities. This structural aspect of the theory under development puts into sharp evidence its wide divergence from the classical scientific conception of reality. Our intensive search failed to discover a dynamical law for the proposed new theory within the mathematical context of the traditional, σ-field-real-valued-measure representation of stochastic processes. We therefore have gone on to a higher level of possible mathematical representation. This particular representation would have collections of eventualities represented by lattices of certain configurations in a unitary space. Closed linear manifolds are among these configurations, and indeed the formalism of quantum mechanics falls into the context of this representation. But the general lattice element in the kinds of lattices being investigated is more complicated than a linear manifold, being heuristically comparable with an acute-angle cone of vectors. These configurations are formulated by means of a ƒlanking relation between linear manifolds. One such lattice has been developed. Others are currently being sought through variations on the notion of flanking. The fact that probabilities present themselves in the unitary space representation-as is familiar from quantum mechanics-as parameters of the relationship between linear manifolds suggests that probability is not a fundamental structural concept. It is consequently indicated that eventualities and acts alone are the basic elements of real structure. Accordingly, the dynamical law must be formulable in terms of these elements only. There is a natural suggestion of what the law might look like. Even at the present early stage of development of the new theory being sought it is possible to cite many pieces of empirical evidence vouching for its exclusive validity. Such evidence is particularly noteworthy in the domain of human behavior.