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General Nonlocal Probability of Arbitrary Order.

Vasily E Tarasov1,2

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|June 28, 2023
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Summary
This summary is machine-generated.

This study introduces a nonlocal generalization of probability using fractional calculus. It defines fractional extensions of probability functions, enabling broader applications in probability theory.

Keywords:
fractional derivativesfractional integralsgeneral fractional calculusnonlocal probabilityprobability theory

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Area of Science:

  • Mathematics
  • Probability Theory
  • Fractional Calculus

Background:

  • Traditional probability theory often assumes local interactions.
  • General Fractional Calculus (GFC) provides tools for nonlocal phenomena.
  • Existing GFC frameworks may limit the scope of nonlocal probability models.

Purpose of the Study:

  • To propose a nonlocal generalization of probability theory.
  • To extend concepts like probability density functions (PDFs) and cumulative distribution functions (CDFs) using fractional calculus.
  • To explore a wider range of nonlocal operators in probability.

Main Methods:

  • Utilizing Luchko's General Fractional Calculus (GFC).
  • Employing the multi-kernel extension of GFC for arbitrary orders (GFC of AO).
  • Defining and analyzing nonlocal and general fractional extensions of probability functions.

Main Results:

  • Successfully defined nonlocal and general fractional (CF) extensions of PDFs and CDFs.
  • Demonstrated the properties of these novel probability distributions.
  • Introduced a framework for considering a wider class of operator kernels and nonlocality.

Conclusions:

  • The proposed GFC-based approach offers a powerful nonlocal generalization of probability.
  • This framework expands the applicability of probability theory to more complex systems with nonlocality.
  • The multi-kernel GFC further enhances the flexibility and scope of these nonlocal probability models.