Partial Differential Equations and Diffusion Processes
Tue 14-15 Room 342 & Thu 10 - 12 / Room 658 Huxley
The exam will take place on --- from
--- to --- in room ---
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 C2
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
- The Harnack inequality and the Liouville
- Convergence of sequences of harmonic
Part II: Sub- and superharmonic functions and the
- C2 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
- 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
- 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
- (optional) The reduced function and the
- (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.
Recommended reading on the contents of the
- ***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
- Chung K.L. "Lectures
from Markov processes to Brownian motion" Springer
1991. 519.217 [Chapter 4 contains topics
- 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
- Treves F. "Basic linear
partial differential equations", Academic Press,
1975. 517.956 TRE [Section 10 contains topics
- 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
Recommended reading on advanced calculus
- 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
- 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
Recommended reading on probability 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
- 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
- Shiryaev A.N. "Probability".
[Chapter II contains foundations of probability
theory. Chapter VIII contains Markov chains].
Click TO GO TO Pss & solns
History of Mathematics & Physics
* Brown, Robert (1773-1858)
Albert (1879-1955) & Smoluchowski,
, 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,
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
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
Bachelier´s life and work: An article published in
Mathematical Finance (Vol. 10 No. 3, p. 339-353 (2000)) .
Rich Bass Lecture Notes
Dennis Silverman , Solution
of the Black Scholes Equation using the Green's Function of the
Explanation of Brownian Motion
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