Heat Kernel Behavior at Infinity on Graphs

• Applicant: Dr. Christian Rose
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 540199605

Distribution laws of diffusion processes are modeled on spaces equipped with a natural Laplace operator acting on functions via the heat equation. Laplace operators encode the local geometry of the space and small changes of the local geometry affect the distribution law of the process. On the other hand, from the globally defined diffusion process one can study global geometric information of the underlying space. The fundamental aspect of the present proposal is the study of deep connections between geometry and analytic phenomena modeled by the heat equation. While this is a classical topic in the smooth setting of manifolds, there are important subjects on graphs which are only marginally understood. In particular, this concerns very irregular situations such as graphs with unbounded vertex degree. Our aim is to derive new insights in the properties of heat kernels on such graphs. Moreover, these observations allow us to study important large scale properties of the underlying structures. The heat kernel is the positive fundamental solution of the heat equation. It naturally encodes geometric information analytically, and thus becomes a very interesting and important object. In fact, the local behavior of the heat kernel is encoded in the local geometry while the global behavior reflects the global geometry. We focus on the following four projects: (i) Characterization of Gaußian upper bounds (ii) Characterization of full Gaußian bounds
 (iii) Elliptic versus parabolic Harnack inequalities (iv) Spaces of harmonic and ancient caloric functions. The relevance of studying the behavior of the heat kernel is not only justified by its mathematical beauty but has been proven by the enormous amount of applications in various fields of mathematics. It is a central topic in partial differential equations and by its direct probabilistic interpretation it is as im- portant in probability and statistics as well. The heat flow is also a major tool in geometry and topology as it is in spectral theory. Moreover, heat kernel estimates are relevant in numerical analysis, e.g., in the construction of wavelets. More specifically, the research topic aims to understand heat kernel estimates beyond the restrictive geometric limitations which are imposed classically. This is not only relevant from a pure mathematical point of view but also for applications to real life networks which often come with few vertices of very high degree (hubs) and a majority of low degree vertices.

• DFG Programme: Research Grants