Math 606: Topics in Stochastic Processes, Summer 2022

Math 606, Summer 2022.
Topics in Stochastic Processes (054–39482R)
Gaussian Processes
Class meetings: MW, 9:30am-12:30pm, VHE 210.

Information on this and related pages changes frequently.

Instructor: Sergey Lototsky.

Office: KAP 248D.
Phone: 213 740 2389
E-mail: lototsky usc edu.

URL: https://dornsife.usc.edu/sergey-lototsky/

Office Hours: MW before and after the class. Appointments at other time are welcome.

Course objective: To learn the foundations of the theory of Gaussian processes. More specifically, a Gaussian process X = X(t), t ∈ [0, T], is a collection of random variables such that, for every finite set {t_1, . . . , t_n} ⊂ [0, T], the random vector (X(t_1), . . . ,X(t_n)) is Gaussian. Such a process has a number of remarkable properties. The story becomes even more interesting once we allow the domain of X to be an arbitrary set and allow X to take values in a locally convex linear topological space. In this class, we will use probabilistic and analytical tools to understand basic results in the theory of Gaussian processes. The topics will include

• Main examples [Brownian motion, bridge, and sheet; Ornstein-Uhlenbeck process,
  fractional Brownian motion, Gaussian free field, etc.]
• Various representations of the Gaussian processes;
• Basic properties of sample paths (continuity, Borel-TIS inequality, large and
  small deviations, etc.);
• Spectral theory;
• Gaussian measures on a locally convex linear topological space;
• Cameron-Martin theorem.

Course work: Class participation, homework assignments, final presentation.

Official grading scheme: 20% class participation, 40% homework assignments, 40% final presentation.

Main reference: Mikhail Lifshits, Lectures on Gaussian processes. Springer Briefs in Mathematics. Springer, Heidelberg, 2012.  x+121 pp. The book is available in electronic form from the USC Libraries

Other references

• Mishura, Yuliya; Zili, Mounir Stochastic analysis of mixed fractional Gaussian processes. ISTE Press, London; Elsevier Ltd, Oxford, 2018. xvi+194 pp.
• Bovier, Anton Gaussian processes on trees: From spin glasses to branching Brownian motion. Cambridge University Press, Cambridge, 2017. x+200 pp.
• Adler, Robert J. The geometry of random fields. Reprint of the 1981 original. SIAM, Philadelphia, PA, 2010. xxi+280 pp.
• Adler, Robert J.; Taylor, Jonathan E. Random fields and geometry.Springer, New York, 2007. xviii+448 pp.
• Marcus, Michael B.; Rosen, Jay Markov processes, Gaussian processes, and local times. Cambridge University Press, Cambridge, 2006. x+620 pp.
• Piterbarg, Vladimir I. Asymptotic methods in the theory of Gaussian processes and fields. Translated from the Russian by V. V. Piterbarg. Revised by the author. American Mathematical Society, Providence, RI, 1996. xii+206 pp.
• Lifshits, M. A. Gaussian random functions. Kluwer Academic Publishers, Dordrecht, 1995. xii+333 pp.
• Yurinsky, Vadim Sums and Gaussian vectors. Lecture Notes in Mathematics, 1617. Springer-Verlag, Berlin, 1995. xii+305 pp.
• Hida, Takeyuki; Hitsuda, Masuyuki Gaussian processes. American Mathematical Society, Providence, RI, 1993. xvi+183 pp.
• Adler, Robert J. An introduction to continuity, extrema, and related topics for general Gaussian processes. Institute of Mathematical Statistics, Hayward, CA, 1990. x+160 pp.
• Ibragimov, Ilʹdar Abdullovich; Rozanov, Y. A. Gaussian random processes. Springer-Verlag, New York-Berlin, 1978. x+275 pp.
• Dym, H.; McKean, H. P. Gaussian processes, function theory, and the inverse spectral problem.  Academic Press, New York-London, 1976. xi+335 pp.

• Urbina-Romero, Wilfredo Gaussian harmonic analysis.  Springer, Cham, 2019. xix+477 pp.
• Hu, Yaozhong Analysis on Gaussian spaces. World Scientific, 2017. xi+470 pp.
• Mandrekar, Vidyadhar S.; Gawarecki, Leszek Stochastic analysis for Gaussian random processes and fields. CRC Press, Boca Raton, FL, 2016. xxii+179 pp.
• Bogachev, Vladimir I. Gaussian measures.  American Mathematical Society, Providence, RI, 1998. xii+433 pp.
• Janson, Svante Gaussian Hilbert spaces. Cambridge University Press, Cambridge, 1997. x+340 pp.
• Kuo, Hui Hsiung Gaussian measures in Banach spaces. Lecture Notes in Mathematics, Vol. 463. Springer-Verlag, Berlin-New York, 1975. vi+224 pp.

• Kocijan, Juš Modelling and control of dynamic systems using Gaussian process models. Springer, Cham, 2016. xvi+267 pp.
• Gualtierotti, Antonio F. Detection of random signals in dependent Gaussian noise. Springer, Cham, 2015. xxxiv+1176 pp.
• Shi, Jian Qing; Choi, Taeryon Gaussian process regression analysis for functional data. CRC Press, Boca Raton, FL, 2011. xx+196 pp.
• Rasmussen, Carl Edward; Williams, Christopher K. I. Gaussian processes for machine learning.  MIT Press, Cambridge, MA, 2006. xviii+248 pp.
• Rue, Håvard; Held, Leonhard Gaussian Markov random fields. Chapman & Hall/CRC, Boca Raton, FL, 2005. xii+263 pp.
• Rosenblatt, Murray Gaussian and non-Gaussian linear time series and random fields. Springer-Verlag, New York, 2000. xiv+246 pp.

• Talagrand, Michel Upper and lower bounds for stochastic processes. Second edition. Springer, Cham, 2021. xviii+726 pp.
• Ledoux, Michel; Talagrand, Michel Probability in Banach spaces: Isoperimetry and processes. Reprint of the 1991 edition. Springer-Verlag, Berlin, 2011. xii+480 pp.
• Ledoux, Michel The concentration of measure phenomenon. American Mathematical Society, Providence, RI, 2001. x+181 pp.

An example of a book review from Math reviews [Edition 1, Edition 2] and from the Bulletin of the AMS

The course file, including homework problems. Aim at two problems per week.

My notes:

• A time line
• Gaussian objects
• Pictures of Brownian sheet, The Kiefer field, Brownian bridge twice, GFF; the Matlab code
• Mercer’s theorem
• Abstract Wiener space
• RKHS
• Stochastic analysis in continuous time
• A summary of Brownian motion
• A summary of Gaussian inequalities
• A summary of large deviations
• A summary of SODEs
• A summary of the Cameron-Martin-Girsanov theorem and related results
• The Weierstrass Approximation Theorem
• Notes about Kalman filter and related topics

Other notes

• Lecture notes for math 547 (statistical learning theory) by Steven Heilman
• Donsker’s Theorem [lecture notes by Davar Khoshnevisan]
• An article about  sampling theorem (in signal processing)
• The original paper by James Mercer containing his theorem
• The original paper by C. Borell
• The Brunn-Minkowski inequality
• Geometric Gaussian inequalities
• A survey of Kalman filter
• In nature: GFF, Corner Growth (Iceland), Corner Growth (Armenia)

Our progress

May 18: Various characterizations of a Gaussian vector.

May 23: Examples of Gaussian processes.

May 25: Spectral and KL representations.

May 30: Memorial Day, no class.

June 1: Abstract Wiener space.

June 6: RKHS; Markov property.

June 8: An overview of Gaussian inequalities.

June 13: Gaussian inequalities.

June 15: Large and small deviations.

June 20: Large deviations and applications.

June 22: The Cameron-Martin-Girsanov theorem.

June 27: Filtering in general and Kalman filter in particular.

June 29: The final discussion.