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arXiv · 2402.02493

Indicator Random Processes and its Application for Modeling Open Stochastic Systems

Valery Doobko·Elena Karachanskaya
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Abstract

The authors present a method of indicator random processes, applicable to constructing models of jump processes associated with diffusion process. Indicator random processes are processes that take only two values: 1 and 0, in accordance with some probabilistic laws. It is shown that the indicator random process is invariant when reduced to an arbitrary positive degree. Equations with random coefficients used in modeling dynamic systems, when applying the method of indicator random processes, can take into account the possibility of adaptation to external changes, including random ones, in order to preserve indicators important for the existence of the system, which can be continuous or discrete. In the case of indicator random processes, defined as functions of the Poisson process, equations for dynamic processes in a media with abruptly changing properties are constructed and studied. To study the capabilities of the proposed method, dynamic models of the diffusion process in media with delay centers and diffusion processes during transitions by switching from one subspace to another were studied. For these models, equations for characteristic functions are constructed. Using the method of indicator random processes, a characteristic function for the Kac model was constructed. It is shown that in the case of dependence of the indicator random process on the Poisson process, the equation for the characteristic function corresponds to the telegraph equation. This result coincides with the result of Kac.

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Valery Doobko, Elena Karachanskaya. 2024-02-04. Indicator Random Processes and its Application for Modeling Open Stochastic Systems. https://arxiv.org/abs/2402.02493

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