Random walk boundaries: their entropies and connections with Hecke pairs
Doctoral thesis, 2020
We present three papers in non-singular dynamics concerning boundaries of random walks on locally compact, second countable groups. One common theme is entropy. Paper II and III are concerned with boundary entropy spectra, while Paper I studies topological properties of entropy. In Paper II we moreover establish a technique to relate random walks on locally profinite groups to random walks on dense discrete subgroups, by the concept of Hecke pairs, which is also used in Paper III.
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.
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.
Poisson boundary
Hecke pairs
Non-singular dynamical systems
Schlichting completion
random walks on groups
non-monotone sequences of $\sigma$-algebras
Furstenberg entropy
Author
Hanna Oppelmayer
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
Randomness is a crucial concept of life. From evolution to climate, from culture to finance, probabilities play an important role - in everyday life as well as in long term predictions. The thesis in hand studies dynamics that follow random rules on an abstract structure. An example of such a process is a random walk on a tree: At each junction we have to make up our minds which branch to climb next and we will do this in accordance with given probabilities. Questions asked in this research area concern so-called boundaries of random walks, that is the picture we see when time goes to infinity. How do the paths of all the people look like and how likely are they? Can these infinite paths be clustered such that the starting point does not matter (for example by having a zig-zag in the path)? What happens if there is a drift in a certain direction?
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.
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.
Subject Categories
Mathematics
ISBN
978-91-7905-370-3
Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4837
Publisher
Chalmers