The aim of the research project is to introduce time-fractional evolution equations with complex spatial variables as well as to study the associated stochastic processes. The complex versions of the classical evolution equations (such as heat and wave equations), obtained by ''complexifying'' the spatial variable only (and keeping the time variable real). This ''complexification'' is studied by two different methods, which produce different equations: first, one complexifies the spatial variable in the corresponding semigroups of operators and secondly, one complexifies the spatial variable in the corresponding evolution equation. We will replace the time derivative with the (generalized) fractional Caputo operator in the heat equations with complex spatial variables. Fractional calculus has attracted lots the attention of several researchers working in several fields including mathematics, physics, chemistry, engineering, hydrology and even finance and social sciences. The time-fractional diffusion equation has been widely used to model the anomalous diffusions exhibiting subdiffusive behavior. Hence, we will study the probabilistic solutions of the so-introduced fractional complex heat equations. In particular, a deep analysis leads to discover a relationship between the above solutions and the subordinated Markov process and anomalous diffusions in the complex plane. Particularly interesting is the interpretation of such random processes as Brownian-type motion on a circle.
Since the wide spectrum of our research, we will interact with external collaborators having specific expertise.
It is worthwhile to observe that (up to our knowledge) this research project is the first attempt to develop fractional stochastic models in the complex space.
