Multivariate smoothing equations with coefficients from the general linear group

• Applicants: Professor Dr. Matthias Meiners; Professor Dr. Sebastian Mentemeier
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 555156216

Subject of the proposal are smoothing equations for random vectors. A random vector solves a smoothing equation if it has the same distribution as a weighted sum of independent copies of itself plus a random shift. The weights may themselves be random matrices. Smoothing equations, also known as the fixed-point equations of the smoothing transformation, are solved by various limits of quantities of interest in models of applied probability and statistical physics. The goal of the project is to characterize the set of solutions, and to determine properties of solutions such as the geometry of the support, the existence of densities, decay and smoothness of these provided they exist, and the asymptotic behavior of the tail probabilities of solutions. Methods come from the fields of branching processes, Choquet-Deny theory, Fourier transformation, functional equations in probability theory, infinitely divisible distributions, martingale theory, Markov chains with general state space, Markov renewal theory, (Poisson) point processes, products of random matrices, random walks and random walks conditioned to stay nonnegative, and random walks on non-commutative groups.

• DFG Programme: Research Grants
• International Connection: China
• Cooperation Partner: Professor Dr. Hui Xiao