Final Activity Report Summary - FRACTAL-LIKE SPACES (Analysis on fractal-like spaces)
There were four principal parts of project. The first part dealt with simple random walk on fractal-like graphs and relations to the geometry of these graphs. For one large class of such graphs it was possible to obtain sharp asymptotic results with respect to time of the return probabilities of the walk. The existence of logarithmic oscillations was proved in almost all cases showing a strong analogy to the behaviour of the volume growth.
The second part of the research project was devoted to the study of the spectrum of the discrete Laplace operator on fractal-like. A remarkable connection of the so-called spectral dimension of infinite symmetric self-similar graphs and combinatorial quantities of finite sections of these graphs was discovered and rigorously proved. In addition, previous work on the structure of the spectrum of the Laplace operator was investigated and somewhat unified.
Heat kernels on metric measure spaces were studied in the third part. Especially, embedding theorems of function spaces associated to the heat kernel and the Laplace operator were investigated. This yielded new aspects of known results and some minor improvements.
Time constraints of the project were too tight, so that only basic work on the last part was possible. This part dealt with jump processes in metric measure spaces. Nevertheless, research on the third and fourth part would be continued, since the fellow accepted a research position in the work group of the scientist in charge.
The second part of the research project was devoted to the study of the spectrum of the discrete Laplace operator on fractal-like. A remarkable connection of the so-called spectral dimension of infinite symmetric self-similar graphs and combinatorial quantities of finite sections of these graphs was discovered and rigorously proved. In addition, previous work on the structure of the spectrum of the Laplace operator was investigated and somewhat unified.
Heat kernels on metric measure spaces were studied in the third part. Especially, embedding theorems of function spaces associated to the heat kernel and the Laplace operator were investigated. This yielded new aspects of known results and some minor improvements.
Time constraints of the project were too tight, so that only basic work on the last part was possible. This part dealt with jump processes in metric measure spaces. Nevertheless, research on the third and fourth part would be continued, since the fellow accepted a research position in the work group of the scientist in charge.