An Application of the S-Functional Calculus to Fractional Diffusion Processes

In this paper we show how the spectral theory based on the notion of S-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the H∞ functional calculus and we use it to define the fractional powers of vector operators. The Fourier law for the propagation of the heat in non homogeneous materials is a vector operator of the formT= e1a(x) ∂x1+ e2b(x) ∂x2+ e3c(x) ∂x3where eℓ, ℓ= 1 , 2 , 3 are orthogonal unit vectors, a, b, c are suitable real valued functions that depend on the space variables x= (x1, x2, x3) and possibly also on time. In this paper we develop a general theory to define the fractional powers of quaternionic operators which contain as a particular case the operator T so we can define the non local version Tα, for α∈ (0 , 1) , of the Fourier law defined by T. Our new mathematical tools open the way to a large class of fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working on fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis.