Homogenization of diffusions with interfaces

• Applicant: Dr. Olga Aryasova, Ph.D.
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 548113512

In many fields of natural sciences one often comes across materials that consists of a homogeneous matter interlaced with very thin layers of different compositions or properties. These can be composite materials like glass wool reinforced with aluminium foil or biological compartments enclosed by thin membranes. The thin layers, also called barriers, membranes or interfaces, can significantly alter the material’s physical properties. In mathematical terms, the characteristics of such materials can be described with the help of elliptic or parabolic partial differential equations with certain boundary conditions on the interfaces. Alternatively, one can follow the stochastic approach and study stochastic differential equations with local time terms that take into account the diffusion’s singular behaviour on the interface. The homogenization problem addressed in this project focuses on analysis of macroscopic limits of such diffusions as the number of interfaces increases while the distances between them decrease. Firstly, we will develop a unified approach to stochastic dynamics across semi-permeable interfaces. Our aim is to establish a general limit theorem for multivariate diffusions with interfaces, considering mild conditions on the regularity of the coefficients. This theorem will encompass a wide range of interface behaviors, such as sliding along the interface or sticking to it. In particular, it will be of utmost importance to carefully address questions regarding the existence and uniqueness of solutions of multivariate stochastic differential equations with local time terms. Furthermore, we consider homogenization along a family of curvilinear interfaces belonging to a foliation of a diffusion’s state space. These interfaces can take various forms, such as spherical or cylindrical interfaces. Additionally, our investigation will encompass the study of media exhibiting randomly perturbed or missing interfaces, as well as periodic media with semi-permeable interfaces. The outcomes of this research will be derived through the utilization of methods of stochastic analysis, specifically incorporating the theory of singular diffusions with local times. We expect that the results of this research can be applied to various physical (in particular, geophysical) and biological models.

• DFG Programme: WBP Position