Diffusion on irregular sets

• Applicant: Professor Dr. Marc Keßeböhmer
• Subject Area: Mathematics
• Term: from 2015 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 281034495

The leading aim of this project is to understand diffusion processes on highly irregular sets. Fractal geometry is a mathematical language and discipline used to describe, study and analyse properties of such structures. A diffusion process models the continuous propagation of a particle or heat conduction in a given medium. The theory of diffusion processes and harmonic structures on homogenous self-similar fractals has been intensively investigated, for example by Barlow, Denker, Hambly, Hattori, Kigami, Lau, Lindstrom, Lapidus and Metz.The results of this project we believe will give a deeper insight into the behaviour of diffusion on highly irregular media in nature. Recent investigations have shown that applications can be found in, for instance, neuronal flows in the brain cortex, oxygen transport in the human lung and gas propagation in rock layers.The first part of the project is to construct diffusion processes on inhomogenous self-similar, self-affine and self-conformal fractals via random walks. The extension of the current known theory to these more general fractals will require a combination of well-established theories and innovative methods, for example, renewal theory, non-stationary multi-type branching processes and thermodynamic formalism. The random walk approach allows for an intuitive and geometric construction yielding new insights, and which is similar to a construction of a Brownian motion on n-dimensional Euclidean space. In our setting, several substantial difficulties arise in that fractals often lack certain regularity conditions, such as symmetry. Some of these difficulties have been overcome on a specific set of self-similar fractals by using an analytic approach due to Kigami, with various further developments by, for instance, Freiberg, Hambly, Strichartz and Teplyaev. In these cases, we are confident that the construction involving random walks will give further information and insight on the behaviour of diffusion processes on fractals and also allow for further generalisations.There is a well-known correspondence between diffusion processes and Laplacians. In the second part of the project, estimates on the transition density of the diffusion processes we will construct will yield the walk dimension and the spectral dimension of the resulting Laplacian. These different notions of dimension shall then be related to a fractal dimension of the set. Namely, we will establish an Einstein-like relation for a wide class of fractals - currently known for self-similar fractals only.The final objective is to establish a connection of diffusion processes to non-commutative geometry; that is, we will compare the resulting Laplacian to the square of Dirac operators proposed by the principle investigator, Falconer, Hinz, Kelleher, Samuel and Teplayev. Further, a class of KMS-states (a generalisation of Gibbs states) motivated by notions in quantum statistical mechanics, will be investigated.

• DFG Programme: Research Grants