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Related Experiment Videos

Right-shift processes.

N C Severo1

  • 1STATE UNIVERSITY OF NEW YORK AT BUFFALO.

Proceedings of the National Academy of Sciences of the United States of America
|December 1, 1969
PubMed
Summary

This study introduces a novel class of Markov processes for multidimensional finite state spaces. We demonstrate a method to simplify Kolmogorov forward equations into a solvable triangular system.

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Area of Science:

  • Probability Theory
  • Stochastic Processes
  • Mathematical Physics

Background:

  • Markov processes are fundamental in modeling systems with state transitions.
  • Multidimensional state spaces introduce complexity in analyzing these processes.
  • Kolmogorov forward equations describe the time evolution of probabilities in Markov processes.

Purpose of the Study:

  • To introduce a new class of Markov processes with specific transition dynamics.
  • To develop an efficient method for solving the associated Kolmogorov forward equations.
  • To explore the mathematical structure of these processes on multidimensional finite state spaces.

Main Methods:

  • Defining a class of Markov processes with unidirectional, one-unit shifts between coordinates.
  • Formulating the infinitesimal generator and transition rates for this process.
  • Representing the Kolmogorov forward equations in a matrix form.
  • Demonstrating the triangular structure of the resulting system.

Main Results:

  • The proposed Markov processes exhibit a specific type of state transition.
  • The Kolmogorov forward equations for these processes can be systematically reduced.
  • A triangular system representation is achieved, simplifying analytical solutions.
  • This structure facilitates the computation of transition probabilities.

Conclusions:

  • The presented class of Markov processes offers a tractable model for certain dynamic systems.
  • The triangular system representation provides an efficient method for analysis.
  • This work contributes to the understanding of multidimensional stochastic processes.

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