Tue 14-15 Room 342 & Thu 10 - 12 / Room 658 Huxley Bldg

The exam will take place on --- from --- to --- in room ---

**Content**

**Part I: The Laplace operator and harmonic functions.**

- The Laplace operator in mathematics and physics. Examples of harmonic functions.
- The Dirichlet problem. The maximum principle and its applications: the uniqueness for the Dirichlet problem; the comparison principle; the proof of the fundamental theorem of algebra.
- Green's formulae. The fundamental solution
of the Laplace operator. Representation of
*C*^{2 }functions via the fundamental solution. - The Green function. Solving the Dirichlet problem in terms of the Green functions.
- Computing the Green function of a ball. The Poisson formula for a solution of the Dirichlet problem in a ball.
- Local properties of harmonic functions: smoothness, the mean value property, the strong maximum principle.
- The Harnack inequality and the Liouville theorem.
- Convergence of sequences of harmonic functions.

**Part II: Sub- and superharmonic functions and the
Dirichlet problem.**

*C*^{2}and*C*sub- and superharmonic functions. Elementary operations on subharmonic functions. The strong maximum principle for subharmonic functions. The comparison principle.- Perron's solution to the Dirichlet problem in arbitrary domains. The harmonicity of Perron's solution.
- Regular and irregular boundary points. The boundary behaviour of Perron's solution. The ball condition and the cone conditions for regularity.
- Capacity. Wiener's criterion of the regularity of boundary points (outline).

**Part III: The Brownian motion and the heat equation.**

- Outline of construction of the Brownian motion. The transition function as a solution to the heat equation.
- The heat kernel and its basic properties. The heat semi-group. Solving the Cauchy problem for the heat equation.
- The parabolic boundary and the parabolic maximum principle. The uniqueness for the initial-boundary problem. The uniqueness for the Cauchy problem in the class of bounded functions.
- The probability space associated with the Brownian motion. Representation of bounded solutions of the Cauchy problem by using the Brownian motion.
- The Markov property of the Brownian motion. The time shift operator.
- Stopping times. The first entrance and the first exit time. The strong Markov property.
- Solving the Dirichlet problem in unbounded domains by using the first exit time. Probabilistic definition of the boundary regularity. The generalized cone condition.
- (optional) The reduced function and the hitting probability.
- (optional) The recurrence and transience of the Brownian motion in Euclidean spaces.

*Calculus of several variables. Basic
Ordinary Differential Equations. Basic Probability Theory
including the normal distribution. Some acquaintance with Measure
theory and Lebesgue integration.** *

- ***
__Bass R.F.__"Probabilistic techniques in analysis", Spinger, 1995.*** __Chung K.L., Zhao Z.__"From Brownian motion to Schrödinger's equation" Springer 1995.**519.217.5 CHU***[Chapter 1 contains topics III.4-7]*__Chung K.L.__"Lectures from Markov processes to Brownian motion" Springer 1991.**519.217***[Chapter 4 contains topics III.4-7]*__Evans L.C.__"Partial differential equations", AMS, 1998.**517.95 EVA***[Section 2.2 contains topics I.1-8, Section 2.3 contains topic III.2-3]****__Garabedian P.R.__"Partial differential equations", Wiley, 1964.**517.95 GAR***[Section 9.2 contains topics II.1-3]*- ***
__Gilbarg D., Trudinger N.__"Elliptic partial differential equations of second order", Springer 1977+.**517.956.2 GIL***[Chapter 2 contain topics I.1-8 and II.1-4]**** __Petrovskii I.G.__"Partial differential equations", Iliffee books, 1967.**517.958 PET***[Chapter 3 contains topics I.1-8 and II.1-3, Chapter 4 - topics III.2-3]*__Pinsky R.__" Positive harmonic functions and diffusion", Cambridge University Press, 1995.**517.5 PIN**__Stroock D.W.__"Probability theory. An analytic view", Cambridge University Press, 1993.*[Chapter 8 contains topics III.1-7]*__Treves F.__"Basic linear partial differential equations", Academic Press, 1975.**517.956 TRE***[Section 10 contains topics I.1-8]*__Brzezniak Z., Zastawniak T.__"Basic Stochastic Processes"**(Central Lib) 519.217 BRZ***[Topics related to Chapter III]*__Lieb E.H., Loss M.__"Analysis" Graduate Studies in Mathematics Vol 14, AMS 1997.**517 LIE***[Chapters 9&10 contain material related to topics Chapter I and II.1]*

__Fulks W..__"Advanced Calculus",**517.1 FUL***[Chapter 12 contains multiple integrals and the divergence theorem]*__Fikhtengolts G.M.__"The fundamentals of mathematical analysis", v.2.**517 FIK***[Section 3.2 contains the divergence theorem]*__Stroock D.W.__"A concise introduction to the theoryt of integration", World Scientific, 1990.**517.518.22 STR***[Section IV.2 - Integration in polar coordinates, the higher dimensional balls. Section IV.4 - the divergence theorem. Chapters II-III - the Lebesgue integration]*__Taylor A.__"General theory of functions and integration",**517.5 TAY***[Chapter 2 - Sets in Euclidean spaces, compactness, connectedness. Chapter 7 - Iterated integrals, Fubini's theorem]*__Kreyszig E.__"Advanced Engineering Mathematics"**51.74 KRE***[ Quick Reference and Examples : Chapter 7 - Vector Calculus, Green's Theorem, Chapter 11 - Basic PDEs, Chapter 17 - Complex Analysis Applied to Potential Theory]*

__Adams M., Gillemin V.__"Measure theory and probability theory",**517.518.11 ADA***[This book makes emphasis on measure theoretic aspects of probability theory].*__Bauer H.__"Probability theory and elements of measure theory",**519.21 BAU.***[The first part contains a complete account of measure theory and Lebesgue integration, and can be read independent of the second part which is probability theory].*__Durrett R.__"Probability: theory and examples",**519.21 DUR**.*[Appendix contains a concise exposition of measure theory].*__Feller W.__"An introduction to probability theory and its applications", vol.2.**519.21 FEL**.*[Chapter X contains a general theory of Markov processes].*__Grimmett G.R., Strizaker D.R.__"Probability theory and random processes".*[Chapter 13 contains diffusion processes].*__Lamperti J.__"Probability: a survey of the mathematics theory",**519.21 LAM**.*[Chapter 4 contains two constructions of the Brownian motion].*__Shiryaev A.N.__"Probability".*[Chapter II contains foundations of probability theory. Chapter VIII contains Markov chains].*

**History of ****Mathematics**** ****& ****Physics**

* Brown, Robert (1773-1858)

* Bachelier, Louis (1870-1946)

* Einstein, Albert (1879-1955) & Smoluchowski, Marian (1872-1917)

Einstein A
, Ann.d.Physik 17 (1905) 549 ; Untersuchungen uber die Theorie
der Brownschen Bewegung (Ed. R Furth) (Leipzig: Akademische
Verlag Gesellschaft, 1922) [Investigations on the Theory of the
Brownian Movement (London: Methuen, 1926) (New York: Dover,
1956)]

Smoluchowski M, Ann. Phys. (Leipzig) 21 756 (1906); in Boltzmann
Festschrift (Leipzig: Barth, 1904) p. 627; Ann. Phys. (Leipzig)
25 205 (1908); Phil. Mag. 23 165 (1912); Bull. Ac. Sci. Cracovie,
Classe Sci. Math. Nat. (1911) p. 493

* Laplace, Pierre-Simon (1749-1827)

* Wiener, Norbert (1894-1964)

.

__Some Articles__

Murrad S. Taqqu , *Bachelier
and His Times : Conversation with Richard Bru**, (An
article published in Mathematical Finance - Bachelier Congress
2000, H. Geman, D. Madan, S.R. Pliska, T. Vorst (Eds.), Springer
(2001)) *

*Louis
Bachelier´s life and work**: An article published in
Mathematical Finance (Vol. 10 No. 3, p. 339-353 (2000)) .*

__Some sites__

Dennis Silverman , Solution
of the Black Scholes Equation using the Green's Function of the
Diffusion Equation

__Some Applets__

* Brownian Motion (again)

* Einstein's Explanation of Brownian Motion

__Societies__

:The Bachelier Finance Society

:

:

__You May Like to Visit__**
****
**Pure Mathematics Section

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Zegarlinski****
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