Ricerc@Sapienza

We aim to find new results in Probability, Analysis and Statistics concerning motions on regular and irregular domains with reflection on regular and irregular regions of the domains. By reflection we mean elastic (Robin), slow (sticky) and skew (transmission) reflection. By “irregular domain” we mean “fractal domain” and “domain with fractal boundary or fractal interface”. Our analysis includes the connections between fractional dimension (irregular domains) and fractional calculus, that is equations involving fractional (or non-local) operators. Starting from such results we aim to realize a collection of codes and routines to be included in free and open international projects as Yuima and others. Anomalous behavior can be regarded as the common property for a wide class of phenomena. Such a class includes motions driven by fractional equations and motions on irregular (or non-homogeneous) domains. From our point of view, this respectively agrees with a macroscopic or microscopic analysis. In the macroscopic analysis we bear the fact that a motion exhibits its own anomalous dynamic, that is the motion can be written as a time change of a base process where the governing equation is a fractional partial differential equation. The microscopic analysis aims to relate the anomalous behaviour of the motion with the geometry of the medium (boundaries, interfaces, layers). We pay special attention to reflection on bounded domains and the associated stochastic processes with finite and infinite velocity. We approach the problem from both probabilistic and pure analytical points of view. Thus, we deal with perturbed Dirichlet forms and semigroups, random time changes of stochastic processes and the associated multiplicative (additive) functionals characterizing the time changed semigroups. Then, we move to the numerical and computational procedures needed in the next step. Once the mathematical framework is well defined, we proceed with the implementation of the results and the construction of a package for platforms as R (https://cran.r-project.org/) or Octave (https://www.gnu.org/software/octave/index). Here we consider, among the others, inverse problems (concerning fractional dimension or fractional order of the operators) and statistical problems. We focus on the active participation with the international Yuima Project for R. The routines collected in our packages turn out to be useful to the scientific community and above all, the packages are free and open. The already existing port of Matlab/Octave scripts to R (and from R) brings our project to a more extended audience.