$L^q$ - positivity preservation, and locally defective functions on regular Dirichlet spaces

• Applicant: Professor Dr. Batu Güneysu
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 554640378

A major problem in geometric analysis that has remained open for a long time is the so-called BMS conjecture - named after Braverman, Shubin and Milatovic - from 2002. This conjecture states that a distributional 1-superharmonic and square-integrable function on a suitable space will be automatically non-negative. There is a close connection between this conjecture and the uniqueness of certain quantum systems (through the so-called essential self-adjointness of the Laplace operator). In addition, the question can also be asked for bounded functions instead of square-integrable ones, in which case there is a close connection to the lifetime of the diffusion processes (the so called stochastic completeness of the space). Regularity results for superharmonic or distributional subharmonic functions (so called locally defective functions) are expected to play a central role in solving the conjecture. While the BMS conjecture was only solved last year for Riemannian manifolds and a short time later on so-called local regular Dirichlet spaces, the aim of this project is to investigate the conjecture in its general form, on regular Dirichlet spaces. These spaces contain many important non-local spaces such as fractals. In a further step, the connection between the regularity of bounded locally defective functions and the stochastic completeness of the space will be investigated. Our ultimate goal is to extend the above investigations to certain infinite-dimensional spaces, such as loop spaces. This is the setting of so-called quasi-regular Dirichlet spaces - the most general setting in which these problems can be studied.

• DFG Programme: Research Grants