Random walk boundaries: their entropies and connections with Hecke pairs
Doctoral thesis, 2020
In Paper I we introduce different perspectives and extensions of Furstenberg's entropy and show semi-continuity and continuity results in these contexts. In particular we apply these to upper and lower limits of non-nested sequences of sigma-algebras in the sense of Kudo.
Paper II relates certain random walks on locally profinite groups to random walks on dense discrete subgroups, using a Hecke subgroup, such that the Poisson boundary of the first becomes a boundary of the second one. If the Poisson boundaries of these two walks happen to coincide, then the Hecke subgroup in charge has to be amenable. For some random walks on lamplighter and solvable Baumslag-Solitar groups we obtain that their Poisson boundary is prime and the quasi-regular representation is reducible. Moreover, we find a group such that for any given summable sequence of positive numbers there is a random walk whose boundary entropy spectrum equals the subsum set of this sequence. In particular we obtain a boundary entropy spectrum which is a Cantor set and one which is an interval.
In Paper III we study the boundary entropy spectra of finitely supported, generating random walks on a certain affine group, realizing them as finite subsum sets. We show that the averaged information function of a stationary probability measure does not change when passing to a non-singular, absolutely continuous sigma-finite measure and deduce an entropy formula.
Non-singular dynamical systems
random walks on groups
non-monotone sequences of $\sigma$-algebras
Chalmers, Mathematical Sciences, Analysis and Probability Theory
Michael Björklund, Yair Hartman, Hanna Oppelmayer: Kudo-Continuity of Entropy Functionals
Michael Björklund, Yair Hartman, Hanna Oppelmayer: Random Walks on Dense Subgroups of Locally Compact Groups
Hanna Oppelmayer: Boundary Entropy Spectra as Finite Subsums
In our work we will link random walks which happen on two different structures. We will do this in a way such that the boundaries of the "finer"' structure sit inside the boundaries of the "rougher" structure. Sometimes they will be the same. In these cases we will find examples where the maximal boundary cannot be split up in any reasonable way. Meaning that the paths of all people will be collected in just one space and we cannot build subcategories (like paths with a zig-zag). Exaggeratedly, one might say no matter which path one has taken, the final space will always be the same. At least in these examples.
Another important tool to distinguish different final spaces (boundaries) is to measure their chaotic behaviour. This is done by entropy. We will show that in some situations any chaotic behaviour is possible and give constructions of corresponding random walks. Moreover, we will look at the collection of all entropy numbers of boundaries on a certain structure and show that many interesting shapes can be realized in this way.
Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4837
Chalmers University of Technology
Opponent: Sara Brofferio, Université Paris Sud, France