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.
The analysis of time-fractional diffusion equations has been developed by several authors. The main goals of this research topic are: 1) the study of equations involving non-local pseudo-differential operators; 2) the description of physical phenomena such as anomalous diffusions; 3) the analysis of time-changed Markov processes. Usually the heat-type equations with fractional time derivative have been studied considering the spatial variable in the real (d-dimensional) space.
Up to our knowledge this research project represents the first attempt to study fractional evolution equations with complex spatial variables and their connections with stochastic processes defined in the complex plane. Hence, in our program are addressed new problems of interest in probability, semigroup theory and mathematical physics. More precisely, the main novelties and purposes of our research project are listed below.
1) By exploiting the methods developed in Gal et al. (2008)-(2014), we introduce evolution-type partial differential equations with fractional time derivatives and complex spatial variables. Furthermore, we study the solutions of these equations (i.e. the strongly continuous semigroup operators) and provide their analytic features. Some attention will be devoted to the study of equations with particular kernels in the fractional derivatives (e.g. power functions).
2) The solutions will be probabilistically interpreted as Markov processes on a circle with random time given the inverse of a (continuous) subordinator not necessarily stable. Therefore, it is possible to highlight a bridge between the theory of stochastic processes on the complex space and the theory of the complex heat-type equations. This connection will be extended to other evolution equations such as Laplace and Schrodinger equations which have attracted the attention of many researchers in the field of the probability theory.
3) We are able to define random walks by compositions of random rotations and then we get anomalous diffusion models on a circle. Usually, these random models are useful to describe delayed motions in the real space. Therefore, for the applications in statistical physics, It is interesting to build complex delayed diffusions having a behavior describing trapping phenomena.