Generalised Fractional Evolution Equations of Caputo Type

Abstract: This paper is devoted to the study of generalised time-fractional evolution equations involving Caputo type derivatives. Using analytical methods and probabilistic arguments we obtain well-posedness results and stochastic representations for the solutions. These results encompass known linear and non-linear equations from classical fractional partial differential equations such as the time-space-fractional diffusion equation, as well as their far reaching extensions. \ Meaning is given to a probabilistic generalisation of Mittag-Leffler functions.