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We consider here convolution operators, in the Caputo sense, withnon-singular kernels. We prove that the solutions to some integro-differentialequations with such operators (acting on the space variable) coincide with thetransition densities of a particular class of L\'evy subordinators (i.e.compound Poisson processes with non-negative jumps). We then extend theseresults to the case where the kernels of the operators have random parameters,with given distribution. This assumption allows greater flexibility in thechoice of the kernel's parameters and, consequently, of the jumps' densityfunction.

Stochastic applications of Caputo-type convolution operators with non-singular kernels