LÉVY MEASURES INVOLVING A GENERALIZED FORM OF
FRACTIONAL INTEGRALS
Makoto Maejima
VÍctor Pérez-Abreu
Ken-iti Sato
Abstract: A four-parameter fractional integral transformation of measures on is
introduced and a systematic study of its properties depending on the values of the parameters
is made. Descriptions of its domain, range, and effect on behaviors of measures near or
far from the origin are found. A non-commutative relation with a two-parameter
Upsilon transformation is established in the form for
some and . Then the class of infinitely divisible distributions having Lévy
measures of the form is discussed. It is represented as the class of laws of
improper stochastic integrals with respect to Lévy processes if . For
, it is the class of laws of essentially definable improper stochastic integrals.
2000 AMS Mathematics Subject Classification: 60E07, 60H05.
Keywords and phrases: Fractional integral transformation, stochastic integral mapping,
Upsilon transformation.