A Random Riemann–Liouville Integral Operator

Jorge Sanchez-Ortiz, Omar U. Lopez-Cresencio (), Martin P. Arciga-Alejandre and Francisco J. Ariza-Hernandez ()
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Jorge Sanchez-Ortiz: Facultad de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N, Cd. Universitaria, Chilpancingo P.O. Box 39087, Guerrero, Mexico
Omar U. Lopez-Cresencio: Facultad de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N, Cd. Universitaria, Chilpancingo P.O. Box 39087, Guerrero, Mexico
Martin P. Arciga-Alejandre: Facultad de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N, Cd. Universitaria, Chilpancingo P.O. Box 39087, Guerrero, Mexico
Francisco J. Ariza-Hernandez: Facultad de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N, Cd. Universitaria, Chilpancingo P.O. Box 39087, Guerrero, Mexico

Mathematics, 2025, vol. 13, issue 15, 1-11

Abstract: In this work, we propose a definition of the random fractional Riemann–Liouville integral operator, where the order of integration is given by a random variable. Within the framework of random operator theory, we study this integral with a random kernel and establish results on the measurability of the random Riemann–Liouville integral operator, which we show to be a random endomorphism of L 1 [ a , b ] . Additionally, we derive the semigroup property for these operators as a probabilistic version of the constant-order Riemann–Liouville integral. To illustrate the behavior of this operator, we present two examples involving different random variables acting on specific functions. The sample trajectories and estimated probability density functions of the resulting random integrals are then explored via Monte Carlo simulation.

Keywords: generalized random variable; random operator; fractional operator (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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