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Analytical Methods in Probability Theory

2019 – 2020, X семестр

Информация по курсу

The interaction between probability and analysis, in particular harmonic
analysis, can be traced back to the formative days of both fields. In fact, one
can say that it predates the mathematical «codification» of probability
realized by Kolmogorov’s axioms. Early on this connection was rather implicit,
however in the second half of the last century it was studied and developed (in
works by Burkholder, Gundy, Fefferman, Stein, McKean, Makarov, Banuelos, Peres,
just to name a few) resulting in many groundbreaking advances. In addition to a
new language to describe analytic phenomena they provided an abundance of deep
techniques and ideas that were instrumental in the solution of many problems of
«classical» analysis.

The goal of this course is to elucidate several instances of this relationship
and provide a demonstration of the symbiosis enjoyed by probability and
(harmonic) analysis. This particular field is vast and extensive, and it
continues to grow in many different directions. Therefore the aim is to
concentrate on the most simple (and in a way classical) examples of this kind,
thus, essentially, restricting to the discrete approaches. More specifically,
the topics discussed will cover the representation of functions by dyadic
martingales, the interplay between the behaviour of various maximal functions,
laws of iterated logarithm, and the boundary behaviour of harmonic functions.

The listeners are expected to know some basics of probability and harmonic
analysis, otherwise the course is more or less self-contained.