Il giorno 25 luglio 2022 alle ore 14:00 precise
in aula 3014 del Dipartimento di Matematica e Applicazioni dell’Università di Milano-Bicocca
il prof. Vadim Kaimanovitch (University of Ottawa) terrà un seminario dal titolo:
"Singularity of stationary measures"
Abstract: Standard fixed point theorems imply that any Markov chain on a compact
state space with weak* continuous transition probabilities has a
stationary (invariant) distribution. What can one say about the stationary
measures of "compound" chains obtained by taking a convex combination or
the product of two Markov operators? If these operators have a common
stationary measure, then obviously it is also stationary for the compound
operator, but what if the original stationary measures are just
equivalent? Would the compound chain admit an equivalent stationary
measure as well?
I will answer the latter question in the negative. The corresponding
examples use the boundary processes associated with random walks on the
modular group PSL(2,Z), and amount to exhibiting two step distributions
with the property that their harmonic measures belong to the same measure
class, whereas the harmonic measures of their convex combinations or
convolution happen to be singular. The involved boundary measures are
closely related to the classical constructions of Minkowski and Denjoy.