I KINDLY ASK THOSE INTERESTED IN TAKING THIS COURSE TO SEND ME AN INFORMAL EMAIL MESSAGE ABOUT THEIR INTENTION.
BME STUDENTS: PLEASE ALSO REGISTER FORMALLY AT NEPTUN. (THE COURSE WILL APPEAR SOON ON THE NEPTUN SYSTEM )
ELTE STUDENTS: WELCOME! PLEASE CONSULT WITH YOUR ADVISOR/MENTOR ABOUT FORMALITIES OF CREDIT TRANSFER. USUALLY IT GOES SMOOTHLY.
LECTURES WILL BE HELD ON THURSDAYS, 2-4 PM AT BME-H-306 CLICK HERE FOR LOCATION ON GOOGLE MAPS
DUE TO MY TRAVEL ON THE FIRST WEEK OF THE SEMESTER THE FIRST LECTURE WILL BE HELD ON FEB 26
THOSE STUDENTS WHO TAKE THE COURSE FOR CREDIT: PLEASE LET ME KNOW.
CONDITIONS FOR OBTAINING THE CREDIT: EXTRA READING OF ONE RESEARCH PAPER ASSIGNED BY ME AND WRITING AN ESSAY ABOUT ITS MATHEMATICAL CONTENT. DETAILS WILL BE CLEARLY SET.
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
Random walk in Random Environment (RWRE): motivating examples
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
The reversible case: RW among random conductances, The environment process: reversibility. Martingale approximation: the theorem of Kipnis and Varadhan. Annealed CLT. Quenched vs. annealed.
RW in divergence-free random environment: motivating examples and new aspects of the problem. The environment process: stationarity but no reversibility. Diffusive bounds.
Non-reversible Kipnis-Varadhan theory and its application to RW in div-free RE: annealed CLT.
Harmonic coordinates and quenched CLT.
Superdiffusivity in low dimensions.
Selection of other problems -- if time permits .
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 who miss some of these elements and therefore hesitate about attending should not shy away. The necessary background knowledge can be picked up on the way presuming enough basic mathematical interest and open mind.
Warming up: SSRW and motivating RWRE [last modified 2023-10-18]
One-dimension -- strong laws and limit theorems [last modified 2023-10-22]
Random walk among random conductances (incl. Kipnis Varadhan Thm - reversible setting) [last modified 2023-11-06]
Random walk in divergence-free random environment [last modified 2023-11-14]
Martingale approximation and Kipnis-Varadhan theory -- non-reversible setting [last modified 2023-11-13]
Quenched CLT and Outlook [last modified 2023-12-08]
more to come soon