Kaisa_2012_3_photo by Veikko Somerpuro

Content of the course:

Introduction. Levy's construction of Brownian motion. Levy's modulus of continuity (non-sharp form).
Brownian motion is nowhere differentiable. Law of iterated logarithm.
Stopping times, Blumenthal's 0+1 law, the strong Markov propery.
Zeros and maxima of 1D Brownian motion. Area of 2D Brownian motion.
Harmonic functions, mean value property.
Dirichlet problem, solutions via Brownian motion. Regular domains, Poincare's cone condition. Domains with no solutions to Dirichlet problem.
Poisson problem, Green's function and their probabilistic interpretation.
Transience and recurrence of Brownian motion.
Estimates of hitting probability via Martin energy. Kakutani's theorem.
Brownian motion is a simple curve if and only if the dimension is at least four.
Continuous martingales and optional stopping. Wald lemmas.
Skorokhod embedding.
Donsker invariance principle.
Law of iterated logarithm for random walks. Arcsine laws.

11.12.2017 klo 09:00 - 1.3.2018 klo 23:59


Käyttäjän Konstantin Izyurov kuva

Konstantin Izyurov

Julkaistu, 19.2.2018 klo 14:49

If you have not attended the lectures, but are still planning to pass the course, please contact me as soon as possible.

Käyttäjän Konstantin Izyurov kuva

Konstantin Izyurov

Julkaistu, 5.2.2018 klo 14:42


there will be no lectures or exercise classes on the week from February 12 to February 16. There will be the following replacements:

Tuesday Feb 06, 12:15-14:00, CK111: Lecture by prof. Antti Kupiainen on Minlos' theorem.
Tuesday Feb 20, 10:15-12:00, C122: Lecture;
Friday Feb 23, 10:15-12:00, C122: Exercises.

The material covered in the additional lectures will be self-contained and the rest of the course will not depend on it.

Kind regards,


Tästä osiosta löydät kurssin opetusaikataulun. Tarkista mahdolliset muut aikataulut kuvauksesta.

Ma 15.1.2018
12:15 - 14:00
To 18.1.2018
10:15 - 12:00
Ma 22.1.2018
12:15 - 14:00
To 25.1.2018
10:15 - 12:00
Ma 29.1.2018
12:15 - 14:00
To 1.2.2018
10:15 - 12:00
Ma 5.2.2018
12:15 - 14:00
To 8.2.2018
10:15 - 12:00
Ma 12.2.2018
12:15 - 14:00
To 15.2.2018
10:15 - 12:00
Ma 19.2.2018
12:15 - 14:00
To 22.2.2018
10:15 - 12:00
Ma 26.2.2018
12:15 - 14:00
To 1.3.2018
10:15 - 12:00

Muu opetus

22.01. - 26.02.2018 Ma 10.15-12.00
Konstantin Izyurov
Opetuskieli: englanti



Exercise sheet 1 - due on Monday 22.01

Mark the exercises you have solved in the beginning of the exercise class. (Update Tuesday: a misprint fixed in Exercise 5)

Exercise sheet 2 - due on Monday 29.01

Exerciese sheet 3 - due on Monday 05.02

Exercise sheets 4+5 - due on Monday 19.02


Optional course.

Master's Programme in Mathematics and Statistics is responsible for the course.

The course belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Probability theory I,II, and their prerequisites

A collection of topics in probability and/or stochastic processes

Recommended time/stage of studies for completion: 1. or 2. year

Term/teaching period when the course will be offered: varying

Some of the following: general aspects of stochastic processes, Markov chains and processes, Brownian motion, ergodic theory, large deviations and concentration of measure, geometric probability, integrable probability, point processes and determinantal processes, random graphs


Lectures and exercise classes

Exam and excercises, Course will be graded with grades 1-5

Exam, other methods will be described later