Potential theory on $R^d$ and on graphs
The aim of this course is to present a number of ideas and notions in Potential Theory from several points of view.
First we plan to make a brief introduction to basics, and then discuss some central objects in the area, in the discrete setting and in continuum, switching sometimes between Analytic and Probabilistic language. Such a plan -- if realized in its fullness -- would obviously require several lecture courses, so we will mostly consider the Laplace operator (i.e. Riesz potentials and harmonic functions) and concentrate on several core notions, amongst them being energy and equilibrium distributions (including Frostman's theorems), capacity, Dirichlet problem and regular points, harmonic measure. We will work in the continuous setting in $R^d$, while the discrete models (usually on trees and lattices) will be used as a source of examples, and also to highlight the probabilistic connections. We will mention the differences between the complex plane and R^d, but we will evade the specifics of complex analysis wherever possible.
First we plan to make a brief introduction to basics, and then discuss some central objects in the area, in the discrete setting and in continuum, switching sometimes between Analytic and Probabilistic language. Such a plan -- if realized in its fullness -- would obviously require several lecture courses, so we will mostly consider the Laplace operator (i.e. Riesz potentials and harmonic functions) and concentrate on several core notions, amongst them being energy and equilibrium distributions (including Frostman's theorems), capacity, Dirichlet problem and regular points, harmonic measure. We will work in the continuous setting in $R^d$, while the discrete models (usually on trees and lattices) will be used as a source of examples, and also to highlight the probabilistic connections. We will mention the differences between the complex plane and R^d, but we will evade the specifics of complex analysis wherever possible.
Lecturer
Date
17th September ~ 14th November, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday,Friday | 13:30 - 15:05 | Shuangqing-C546 | ZOOM 12 | 815 762 8413 | BIMSA |
Prerequisite
The listener should be acquainted with basics of real and complex analysis (including measure theory), functional analysis (in particular distributions) and probability, some knowledge of Markov chains, harmonic and subharmonic functions is also recommended. We aim to present the core proofs in a self-contained way, for the rest the references will be provided.
Syllabus
1. Harmonic and subharmonic functions: basic notions, Harnack's inequality, maximum principle, Poisson representation formula, modification and gluing theorems, integrability, smoothing, weak identity, harmonicity and subharmonicity on graphs;
2. Potentials and equilibrium: general definition, Newton, Riesz and Bessel potentials in $R^d$, continuity, maximum principle, energy, polar sets, equilibrium measures, Frostman's theorem, generalized Laplacian, Riesz decomposition theorem;
3. Geometric properties: thinness, Hausdorff content, infinity sets.
4. Capacities: definitions, basic properties, dual definition, capacity estimates, capacity and dimension, some examples.
5. Dirichlet problem for the Laplace operator: formulation, Perron's method, regular boundary points and barriers, criteria for regularity, harmonic measure, equilibrium and harmoinc measures, Green functions, harmonic majorants.
2. Potentials and equilibrium: general definition, Newton, Riesz and Bessel potentials in $R^d$, continuity, maximum principle, energy, polar sets, equilibrium measures, Frostman's theorem, generalized Laplacian, Riesz decomposition theorem;
3. Geometric properties: thinness, Hausdorff content, infinity sets.
4. Capacities: definitions, basic properties, dual definition, capacity estimates, capacity and dimension, some examples.
5. Dirichlet problem for the Laplace operator: formulation, Perron's method, regular boundary points and barriers, criteria for regularity, harmonic measure, equilibrium and harmoinc measures, Green functions, harmonic majorants.
Reference
· David Adams, Lars Inge Hedberg. Function Spaces and Potential Theory. Springer, 1999.
· Thomas Ransford. Potential Theory in the Complex Plane. Cambridge, 1995
· Lester L. Helms. Potential Theory. Springer, 2009
· W.K. Hayman, P.B. Kennedy. Subharmonic functions Vol. 1. London Mathematical Society, 1976
· S. Axler, P. Bourdon, W. Ramey. Harmonic function theory. Springer, 2001
· E.B. Dynkin, A.A. Yushkevich. Markov Processes: Theorems and Problems. New York, 1969
· Russel Lyons, Yuval Peres. Probability on Trees and Networks. Cambridge, 2016
· Richard E. Bass. Probabilistic Techniques in Analysis. Springer, 1995
· Kai Lai Chung. Green, Brown and Probability & Brownian Motion on the Line. World Scientific, 2002
· Joseph A. Doob. Classical Potential Theory and Its Probabilistic Counterpart. Springer, 2001
· Thomas Ransford. Potential Theory in the Complex Plane. Cambridge, 1995
· Lester L. Helms. Potential Theory. Springer, 2009
· W.K. Hayman, P.B. Kennedy. Subharmonic functions Vol. 1. London Mathematical Society, 1976
· S. Axler, P. Bourdon, W. Ramey. Harmonic function theory. Springer, 2001
· E.B. Dynkin, A.A. Yushkevich. Markov Processes: Theorems and Problems. New York, 1969
· Russel Lyons, Yuval Peres. Probability on Trees and Networks. Cambridge, 2016
· Richard E. Bass. Probabilistic Techniques in Analysis. Springer, 1995
· Kai Lai Chung. Green, Brown and Probability & Brownian Motion on the Line. World Scientific, 2002
· Joseph A. Doob. Classical Potential Theory and Its Probabilistic Counterpart. Springer, 2001
Video Public
Yes
Notes Public
Yes
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.