RANDOM WALK IN RANDOM ENVIRONMENT

A TCC COURSE HELD FOR GRADUATE STUDENTS OF MATHEMATICS AT BATH, BRISTOL, IMPERIAL COLLEGE LONDON, OXFORD, SWANSEA AND WARWICK

2023 AUTUMN (OXFORD: MICHAELMAS) TERM 

LECTURER: BÁLINT TÓTH (BRISTOL AND BUDAPEST)

WHEN?

LECTURES HELD ON WEDNESDAYS 10:00-12:00 (UK TIME)

FIRST LECTURE: WEDNESDAY, 11 OCTOBER 2023

LAST LECTURE: WEDNESDAY, 29 NOVEMBER  6 DECEMBER 2023 

WHERE?

THE LECTURES WILL BE HELD ON MICROSOFT TEAMS PLATFORM WITH ACCESS PROVIDED TO REGISTERED ATTENDANTS. 

MESSAGES TO STUDENTS ATTENDING THE COURSE:

8 DECEMBER 2023:

• THANK YOU ALL FOR COMING TO THIS COURSE. 

• DEADLINE FOR  HANDING IN THE ESSAYS: 15 FEB 2024. 

• FORMAT OF ESSAYS: CCA 12-15 PAGES IN STANDARD LATEX IN THE FORMAT OF A RESEARCH PAPER.

SYLLABUS:

1. Survey of main results about simple symmetric random walk on Z^d: Recurrence vs. Transience, Póllya's theorem; Laws of Large Numbers; Central Limit Theorem; Invariance principle

2. Random walk in Random Environment (RWRE): motivating examples

3.  RWRE in one dimension (on Z^1): Recurrence vs transience; Limit theorems in the recurrent case; Stable limit laws in the sub-ballistically transient cases; some other examples and results

4. The reversible case: RW among random conductances, The environment process: reversibility. Martingale approximation: the theorem of Kipnis and Varadhan. Annealed CLT. Quenched vs. annealed. 

5. RW in divergence-free random environment: motivating examples and new aspects of the problem. The environment process: stationarity but no reversibility. Diffusive bounds. 

6. Non-reversible Kipnis-Varadhan theory and its application to RW in div-free RE: annealed CLT. 

7. Harmonic coordinates and quenched CLT. 

8. Selection of other problems -- if time permits . 

PREREQUISITS: 

• • • Fluency in probability theory (including conditional expectation (a la Kolmogorov), Markov chains and Markov processes, laws of large numbers, 0-1 laws, ergodicity, central limit theorems)

    • Fluency in analysis and functional analysis (including  basics of Fourier transform, L^p spaces, analysis in Hilbert space, spectral theorem for self-adjoint operators)

However, those hesitating possible attendants who miss some of these elements should not be shy. The necessary background knowledge can be picked up on the way presuming enough basic mathematical interest and open mind. 

LECTURE NOTES:

1. Warming up: SSRW and motivating RWRE [last modified 2023-10-18]

2. One-dimension -- strong laws and limit theorems [last modified 2023-10-22]

3. Random walk among random conductances (incl. Kipnis Varadhan Thm - reversible setting) [last modified 2023-11-06]

4. Random walk in divergence-free random environment [last modified 2023-11-14]

5. Martingale approximation and Kipnis-Varadhan theory -- non-reversible setting [last modified 2023-11-13]

6. Quenched CLT and Outlook [last modified 2023-12-08]

ASSESSMENT FOR CREDIT:

Those PhD students who take this unit for credit will be assessed in the following way: I will provide some extra material for reading. This will be either a research paper or some theoretical part not fully covered in class. The student will be asked to read and understand the material and to write an essay of cca 15 pages about it, in the style and format of a research paper. 

Timing:  The reading material will be provided towards the end of the TCC term (by end of November or early December). Deadline for handing in the essay sometime in the winter break (mid- or end of January). 

Please let me know whether you take this unit for credit or for audit only.