Self-similar Cauchy problems and generalized Mittag-Leffler functions

Abstract

By observing that the fractional Caputo derivative can be expressed in terms of a multiplicative convolution operator, we introduce and study a class of such operators which also have the same self-similarity property as the Caputo derivative. We proceed by identifying a subclass which is in bijection with the set of Bernstein functions and we provide several representations of their eigenfunctions, expressed in terms of the corresponding Bernstein function, that generalize the Mittag-Leffler function. Each eigenfunction turns out to be the Laplace transform of the right-inverse of a non-decreasing self-similar Markov process associated via the so-called Lamperti mapping to this Bernstein function. Resorting to spectral theoretical arguments, we investigate the generalized Cauchy problems, defined with these self-similar multiplicative convolution operators. In particular, we provide both a stochastic representation, expressed in terms of these inverse processes and an explicit representation, given in terms of the generalized Mittag-Leffler functions, of the solution of these self-similar Cauchy problems.

References in corpus (7)

• Fractional Cauchy problems on bounded domains
• Distributed-order fractional Cauchy problems on bounded domains
• Space-time fractional diffusion on bounded domains
• Time fractional equations and probabilistic representation
• On fully mixed and multidimensional extensions of the Caputo and Riemann-Liouville derivatives, related Markov processes and fractional differential equations
• Relaxation patterns and semi-Markov dynamics
• Generalised Fractional Evolution Equations of Caputo Type