**Introduction**

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 by many famous researchers (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, the boundary behaviour of harmonic functions, in particular the properties of the harmonic measure.

The course is divided into two parts. In this one we use the "harmonic function-dyadic martingale" relation to study various boundary properties of harmonic functions and the boundary behaviour of harmonic measure.

**Syllabus**

i. Introduction: recalling harmonic functions, dyadic martingales and approximations of the former by the latter.

ii. Harmonic measure on the plane and in higher dimensions.

iii. Hausdorff dimension of the harmonic measure I, history up to Makarov.

iv. Hausdorff dimension of the harmonic measure II, Makarov's theorem.

v. Hausdorff dimension of the harmonic measure III, Bourgain's upper estimate.

vi. Hausdorff dimension of the harmonic measure IV, Wolff's snowflakes.

vii. Banuelos-Moore LIL for harmonic functions: upper estimate.

viii. Banuelos-Moore LIL for harmonic functions: lower estimate and some open problems.

ix. Bourgain's theorem on the radial variation and variations thereof.

**Reference**

[1] N. Arcozzi, N. Chalmoukis, M. Levi, P. Mozolyako. Two-weight dyadic Hardy's inequalities. https://arxiv.org/abs/2110.05450

[2] R. Banuelos, C. N. Moore. Probabilistic behavior of harmonic functions. Birkh auser, Basel-Boston-Berlin (1999)

[3] 3C.J. Bishop. Harmonic Measure: Algortihms and Applications. Proc. Int. Cong. Math. -- 2018, Rio de Janeiro, Vol.2 (2018).

[4] J. Bourgain. On the Hausdorff dimension of harmonic measure in higher dimensions. Invent. Math., 87, 477-483.

[5] J. Bourgain. On the radial variation of bounded analytic functions on the disk, Duke Math. J. 69 (1993), no. 3, 671 682.

[6] A. Canton, J.L. Fernandez, D. Pestana, J.M. Rodri guez. On harmonic functions on trees, Potential Analysis, 15 (2001), 199-244.

[7] I. Daubechies, Ten lectures on wavelets, SIAM, Philadelphia, PA, 1992.

[8] K.S. Eikrem, E. Malinnikova, P. Mozolyako. Wavelet characterization of growth spaces of harmonic functions, Journal d'Analyze Math ematique 122 (2014), no. 1, 87 -111.

[9] J. L. Fernandez, J. Heinonen, J. G. Llorente, Asymptotic values of subharmonic functions, Proc. London Math. Soc. 73.2 (1996), no. 3, 404 430.

[10] J.B. Garnett, D.E. Marshall. Harmonic measure. Cambridge University Press, (2005), 571 pp.

[11] V.P. Havin, P.A. Mozolyako, Boundedness of variation of a positive harmonic function along the normals to the boundary. Algebra and Analysis 28 (2016), no. 3, 67–110

[12] J.G. Llorente, Boundary values of harmonic Bloch functions in Lipschitz domains: a martingale approach, Potential Analysis, 9, 229-260 (1998)

[13] J.G. Llorente. Discrete martingales and applications to analysis. Univ. of Jyv\"{a}skyl\"{a} report, (2002).

[14] N.G. Makarov, Probability methods in the theory of conformal mapping, Algebra i Analiz, 3-59 (1989), (Russian) [Engl. transl. Leningrad Math. J., 1, (1990)]

[15] Y. Meyer, Wavelets and Operators, 225 pp. Cambridge University Press, Cambridge, (1992)

[16] T. H. Wolff, Counterexamples with harmonic gradients in R3, pp. 321–384 in Essays on Fourier analysis in honor of Elias M. Stein (Princeton, 1991), Princeton Math. Ser. 42, Princeton Univ. Press, Princeton, NJ, 1995

**Lecturer Intro**

Pavel Mozolyako is an associate professor at St. Petersburg State University. He leads PhD program in mathematics at the department of Mathematics and Computer Science. He got his PhD degree in 2009, at St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences. He was a postdoc at Norwegian University of Science and Technology, University of Bologna, and a visiting professor at Michigan State University. His research considers mostly boundary behaviour of harmonic functions and discrete models in potential theory.