2024/12/20 by T.G. Pace, Pace, Tristan, Gordan Žitković +1
Mathematics · Biochemistry, Genetics and Molecular Biology · #Point processes and geometric inequalities #Diffusion and Search Dynamics
paper · pdf · doi:10.48550/arxiv.2412.15999
We consider a sequence of Hawkes processes whose excitation measures may depend on the generation, and study its scaling limits in the near-unstable limiting regime. The limiting random measures, characterized via a nonlinear convolutional equation, form a family parameterized by a pair consisting of a locally finite measure and a geometrically infinitely divisible probability distribution on the positive real line. These measures can be interpreted as generalizations of the Feller diffusion and fractional Feller (CIR) processes, but also allow for a "driving noise" associated with general L' evy-type operators of order at most 1, including fractional derivatives of any order α>0 (formally corresponding to possibly negative Hurst parameters).