## Partial Differential Equations and Diffusion Processes 2005/06

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

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

Content

### Syllabus

Part I: The Laplace operator and harmonic functions.

1. The Laplace operator in mathematics and physics. Examples of harmonic functions.
2. 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.
3. Green's formulae. The fundamental solution of the Laplace operator. Representation of C2 functions via the fundamental solution.
4. The Green function. Solving the Dirichlet problem in terms of  the Green functions.
5. Computing the Green function of a ball. The Poisson formula for a solution of the Dirichlet problem in a ball.
6. Local properties of harmonic functions: smoothness, the mean value property, the strong maximum principle.
7. The Harnack inequality and the Liouville theorem.
8. Convergence of sequences of harmonic functions.

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

1. C2 and C sub- and superharmonic functions. Elementary operations on subharmonic functions. The strong maximum principle for subharmonic functions. The comparison principle.
2. Perron's solution to the Dirichlet problem in arbitrary domains. The harmonicity  of Perron's solution.
3. Regular and irregular boundary points. The boundary behaviour of  Perron's solution. The ball condition and the cone conditions for regularity.
4. Capacity. Wiener's criterion of the regularity of boundary points (outline).

Part III: The Brownian motion and the heat equation.

1. Outline of construction of the Brownian motion. The transition function as a solution to the heat equation.
2. The heat kernel and its basic properties.  The heat semi-group. Solving the Cauchy problem for the heat equation.
3. 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.
4. The probability space associated with the Brownian motion. Representation of bounded solutions of the Cauchy problem by using the Brownian motion.
5. The Markov property of the Brownian motion. The time shift operator.
6. Stopping times. The first entrance and the first exit time.  The strong Markov property.
7. Solving the Dirichlet problem in unbounded domains by using the first exit time. Probabilistic definition of the boundary regularity. The generalized cone condition.
8. (optional) The reduced function and the hitting probability.
9. (optional) The recurrence and transience of the Brownian motion in Euclidean spaces.

### Prerequisites

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

### Bibliography

##### Recommended reading on the contents of the course
1. ***Bass R.F. "Probabilistic techniques in analysis", Spinger, 1995.***
2. 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]
3. Chung K.L.  "Lectures from Markov processes to Brownian motion" Springer 1991. 519.217 [Chapter 4 contains topics III.4-7]
4. 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]***
5. Garabedian P.R. "Partial differential equations", Wiley, 1964. 517.95 GAR [Section 9.2 contains topics II.1-3]
6. ***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]***
7. 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]
8. Pinsky R. " Positive harmonic functions and diffusion", Cambridge University Press, 1995. 517.5 PIN
9. Stroock D.W. "Probability theory. An analytic view",  Cambridge University Press, 1993. [Chapter 8 contains topics III.1-7]
10. Treves F. "Basic linear partial differential equations", Academic Press, 1975. 517.956 TRE [Section 10 contains topics I.1-8]
11. Brzezniak Z., Zastawniak T. "Basic Stochastic Processes" (Central Lib) 519.217 BRZ [Topics related to Chapter III]
12. 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]
##### Recommended reading on advanced calculus
1. Fulks W.. "Advanced Calculus",  517.1 FUL [Chapter 12 contains multiple integrals and the divergence theorem]
2. Fikhtengolts G.M.  "The fundamentals of mathematical analysis", v.2. 517 FIK [Section 3.2 contains the divergence theorem]
3. 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]
4. 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]
5. 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]
##### Recommended reading on probability theory
1. Adams M., Gillemin V. "Measure theory and probability theory", 517.518.11 ADA [This book makes emphasis on measure theoretic aspects of probability theory].
2. 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].
3. Durrett R. "Probability: theory and examples",  519.21 DUR. [Appendix contains a concise exposition of measure theory].
4. Feller W. "An introduction to probability theory and its applications", vol.2.  519.21 FEL. [Chapter X contains a general theory of Markov processes].
5. Grimmett G.R., Strizaker D.R. "Probability theory and random processes". [Chapter 13 contains diffusion processes].
6. Lamperti J. "Probability: a survey of the mathematics theory", 519.21 LAM. [Chapter 4 contains two constructions of the Brownian motion].
7. Shiryaev A.N. "Probability". [Chapter II contains foundations of probability theory. Chapter VIII contains Markov chains].

GOTO Problem Sets

Click TO GO TO Pss & solns

SOME LINKS

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

Some Applets

* Brownian Motion (again)

Societies

:

:

You May Like to Visit
Pure Mathematics Section

Analysis/Probability Seminar

Page by Boguslaw Zegarlinski : Comments, Remarks and Questions Welcome !