Path integrals of (planar) Brownian motion and related processes

• Applicant: Professor Dr. Leif Döring
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 526069380

Path integrals of stochastic processes have been a central piece in probability theory for decades due to their importance in the study of stochastic differential equations, branching processes, and most importantly to better understand the path behavior of stochastic processes through their occupation times. The proposed project deals with several questions: compute moments and identify exact distributions of occupation times in orthants and cones of d-dimensional Brownian motion and related processes, as well as the time spent positive by one-dimensional Gaussian processes and exchangeable processes. Furthermore, we will characterise finiteness of path integrals for particular corner cases such as the Cauchy process, a question that will settle a remaining open question of SDEs with jumps. To achieve these goals we will trace out a new, simple, but potentially effective approach with minimal assumptions on the underlying process. Our method uses a surprising (simple) connection between occupation times and persistence probabilities of sums of random variables, angles of random cones in stochastic geometry, and orthant probabilities of multidimensional Gaussian random variables. This allows us to study properties of occupation times that have been out of reach so far.

• DFG Programme: Research Grants