Week 25.03.2024 – 30.03.2024

We discuss necessary and sufficient criteria for certain Fourier multiplication operators to satisfy the Liouville property (bounded harmonic functions are a.s. constant) and the local continuation property (bounded functions, that are harmonic and identically zero on a domain, are a.s. zero on the whole space). Since the operators generate stochastic processes, there is also a probabilistic interpretation of these findings.

The Wiener-Hopf factorisation of a Lévy process has two forms. The first describes how the process makes new maxima and minima, by decomposing it into two so-called 'ladder processes'. The second expresses its characteristic exponent as the product of two functions related to the ladder processes. Since the latter is analytic in nature, the question naturally arises: is such a decomposition unique? The answer has been known for killed Lévy processes since at least Rogozin's work in 1966, but appears to have remained open in general. We show that, indeed, uniqueness holds in all cases. This gives a solid foundation to the 'theory of friendship', which allows one to construct a Lévy process with known Wiener-Hopf factorisation. The results also hold for random walks. Joint work with Leif Döring (Mannheim), Mladen Savov (Sofia) and Lukas Trottner (Aarhus).