The analysis of nonlocal porous medium equations, i.e. nonlinear diffusion equations with fractional operators, is a recent research topic in the theory of the partial differential equations. Many papers appeared in the last decade are devoted to the study of such equations but only in the functional analysis framework (i.e. existence and uniqueness of the solution, hypercontractivity estimates and so on). The classical porous medium equation has been introduced to overcome the paradox of infinte propagation of the heat flow. A special solution is the so-called Kompaneets-Zel'dovich-Barenblatt solution which is a compactly supported parabolic function.
The goal of this research project is to study stochastic models associated to the fractional porous medium equations. Therefore, the heat diffusion equations with spatial pseudo-differential operators represent our main object of interest. We will focus our attention on the following issues.
1) We are interested to the study of the nonlocal operators and how they affect the long-range memory of the stochastic processes linked to the fractional equations. The mathematical tools could require the use of the Lévy subordinators.
2) Furthermore, we want introduce reasonable random walks moving in a lattice in order to describe the microscopic dynamic of the fractional porous medium equations.
3) Another research topic concerns the relationship between some nonlocal equations and the random flight processes. These random models describe a particle starting at the origin with a randomly chosen direction and with finite speed. The direction of the particle changes at each collision with some scattered obstacles where a new direction of motions is taken.
Up to our knowledge there are not other research projects on the probabilistic interpretations of the fractional PMEs. The idea of studying nonlinear diffusion equations with fractional operators is a recent challenge in the theory of the partial differential equations. Many papers appeared in the last decade are devoted to the analysis of such equations but only in the pde setting (i.e. existence and uniqueness of the solution, hypercontractivity estimates and so on).
Therefore, in relation to the current knowledge on the subject of the research project, the original contents that we want to investigate are the following:
1) from a probabilistic point of view it is interesting to understand as the fractional nature of the adopted operators affects the long-range memory of the stochastic process linked to the fractional equations. Therefore, a crucial point will be represented by the characterization of the random walks moving in a lattice, in order to microscopically describe the nonlinear and fractional diffusion;
2) connection between random flight models (also called transport processes), which describe the displacements of a particle, with the non-linear and fractional aspects of the studied equations. As part of the probabilistic analysis of the weak solution to the porous media equations, this approach has never been considered before. These random processes seem to recover the properties arising from the analytical study of the compactly supported solutions of the porous medium equation. Indeed, for any fixed time t>0, the transport processes have support in a closed ball;
3) another topic deserving a detailed study is the relationship between subordinator processes and the fractional equations. It is well-known that in the fractional linear case there is a strong interplay between the Lévy subordination and the pseudo-differential operators. Does the same connection hold in the nonlinear fractional case? The reply to this question will be object of our research.
We believe that the previous arguments play a crucial role in understanding the intrinsic random structure of the real phenomena described by the fractional porous medium equations.