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Manjunath Krishnapur

Department of Mathematics, Indian Institute of Science, Bangalore 560 012

Gaussian processes (Autumn 2020)

Online lectures: I record screencasts and share them or put them out in public (if desired, we can also fix a schedule time and then you can listen to the lecture live). Once a week we shall have a meeting on Microsoft teams to take questions and solve problems.


Description: A course in Gaussian processes. At first we shall study basic facts about Gaussian processes - isoperimetric inequality and concentration, comparison inequalities, boundedness and continuity of Gaussian processes, Gaussian series of functions, etc. Later we go to more specialized topics as time permits. For example, smooth Gaussian processes and their nodal sets, in particular expected length and number of nodal sets, persistence probability and other such results from recent papers of many authors. Or basics of multiplicative chaos.

Grading: Solve many exercises throughout the course. There may be an additional midterm and final exam.

Texts and other resources: In no particular order (I may sample material from many places) -
  1. V. I. Bogachev Gaussian Measures, American Mathematical Society, Providence, RI, 1998.
  2. Ofer Zeitouni Gaussian fields: Notes for lectures
  3. Robert Adler and Jonathan Taylor Gaussian Random Fields, Springer, New York, 2007.
  4. (available in IISc via springer link)
  5. Harry Dym and Henry McKean Gaussian Processes, Function Theory, and the Inverse Spectral Problem, Dover Publications Inc. (2008)
  6. Svante Janson Gaussian Hilbert Spaces, Cambridge University Press, Cambridge, 1997
  7. Michel Ledoux and Michel Talagrand Probability in Banach spaces. Isoperimetry and processes, Springer-Verlag, Berlin, 2011.
  8. Boris Tsirelson Lecture notes from a course
  9. Michel Ledoux Isoperimetry and Gaussian analysis, St. Flour lecture notes-1994.
  10. Ramon van Handel Probability in High Dimension



Lecture notes. These combine and update the old notes available at the first version of the course.