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

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Abstract

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 \({H^\infty}\) 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 form

$$T = e_{1} a(x)\partial_{x1} + e_{2} b(x)\partial_{x2} + e_{3} c(x)\partial_{x3}$$

where \({e_{\ell}, {\ell} = 1, 2, 3}\) are orthogonal unit vectors, a, b, c are suitable real valued functions that depend on the space variables \({x = (x_{1}, x_{2}, x_{3})}\) 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 \({T^{\alpha}, {\rm for} \alpha \in (0, 1)}\), 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.

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  1. Politecnico di Milano Dipartimento di Matematica, Via E. Bonardi, 9, 20133, Milano, Italy

    Fabrizio Colombo & Jonathan Gantner

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  1. Fabrizio Colombo
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  2. Jonathan Gantner
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Correspondence to Fabrizio Colombo.

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Colombo, F., Gantner, J. An Application of the S-Functional Calculus to Fractional Diffusion Processes. Milan J. Math. 86, 225–303 (2018). https://doi.org/10.1007/s00032-018-0287-z

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  • DOI: https://doi.org/10.1007/s00032-018-0287-z

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