Subordination formulae for space-time fractional diffusion processes via Mellin convolution

Fundamental solutions of space-time fractional diffusion equations can be interpret as probability density functions. This fact creates a strong link with stochastic processes. Recasting probability density functions in terms of subordination laws has emerged to be important to built up stochastic processes. In particular, for diffusion processes, subordination can be understood as a diffusive process in space, which is called parent process, that depends on a parameter which is also random and depends on time, which is called directing process. Stochastic processes related to fractional diffusion are self-similar processes. The integral representation of the resulting probability density function for self-similar stochastic processes can be related to the convolution integral within the Mellin transform theory. Here, subordination formulae for space-time fractional diffusion are provided. In particular, a noteworthy new formula is derived in the diffusive symmetric case that is spatially driven by the Gaussian density. Future developments of the research on the basis of this new subordination law are discussed.