Relaxation Equations with Stretched Non-local Operators: Renewals and Time-Changed Processes

Abstract We introduce and study renewal processes defined by means of extensions of the standard relaxation equation through “stretched” non-local operators (of order $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and with parameter $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> ). In a first case, we obtain a generalization of the fractional Poisson process, which displays either infinite