Instructor: Antar Bandyopadhyay
Email: antar (at) isid (dot) ac (dot) in
Office: 4.12 A. N. Kolmogorov Bhavan

Class Time: M 17:30 - 18:30, W 14:00 - 16:00
[No class during September 3 - 14, 2008]
[No class on October 29, 2008 (Wednesday) and November 3, 2008 (Monday)]

Instructor's Office Hours: Tu 14:00 - 16:00

Course Duration: August 18 - November 19, 2008.

Course Outline:

• Introduction, basic definitions and notations.
• RWRE for d = 1:
  • Ergodic Theorems, SLLN. Hitting time decomposition, the environment viewed by the particle.
  • Central Limit Theorem.
  • Sinai Model.
• RWRE for d > 1:
  • 0-1 Laws and SLLN.
  • Kalikow's Condition, ballistic walk, CLT.
  • CLT for balanced walk.

Prerequisites:

• Liner Algebra (at the level of Finite Dimensional Vector Spaces, by P. R. Halmos).
• Measure Theoretic Probability (at the level of Probability: Theory and Examples by R. Durrett).
• Functional Analysis (at the level of Functional Analysis by W. Rudin).

References:

• Random Walks in Random Environment, XXXI St. Flour Lecture Notes, 2001, by Ofer Zeitouni.
• A typo list for the published St. Flour Lecture Note on RWRE, by Ofer Zeitouni .
• Random Walks in Random Environment: A Guided Tour, Lecture Notes from Rio de Janeiro, by Ofer Zeitouni.
• Probability: Theory and Examples by R. Durrett.

Additional Papers/References:

• H. Kesten, Sums of stationary sequences can not grow slower than linearly, Proc. AMS 49 (1975), pp. 205 - 211.
  [For a proof of Kesten's Lemma.]

• F. Solomon, Random walks in random environments, Annals Probab. 3 (1975), pp. 1 - 31.
  [The original paper for the ergodic theorems with i.i.d. environment.]

• N. Berger, Limiting velocity of high-dimensional random walk in random environment, Annals Probab. 36(2) (2008), pp. 728 - 738.
  [Very recent work on limiting velocity when d ≥ 5.]

• M. P. W. Zerner, A Non-Ballistic Law of Large Numbers for Random Walks in I.I.D. Random Environment, Elect. Comm. in Probab. 7 (2002), pp. 181 - 187.
  [For the 0-1 Law when the walk does not converge to infinity or negative infinity in the given direction.]

• A. S. Sznitman, Slowdown Estimates and Central Limit Theorem for Random Walks in Random Environment, J. Eur. Math. Soci. 2 (200), pp. 93 - 143.
  [For CLT in dimension d ≥ 2 provided 0-1 Law holds.]

Student Presentations:

• Date: November 10, 2008 (Monday)
  Time: 17:30 - 18:30
  Speaker: Mr. Rajat Subhra Hazra
  Topic: TBA

• Date: November 12, 2008 (Monday)
  Time: 14:00 - 16:00
  Speaker: Ms. Farkhondeh A. Sajadi
  Topic: "A Non-Ballistic Law of Large Numbers for Random Walks in I.I.D. Random Environment" (based on a paper by M. P. W. Zerner, 2002).

• Date: TBA
  Time: TBA
  Speaker: Mr. Koushik Saha
  Topic: TBA

• Date: TBA
  Time: TBA
  Speaker: Mr. Arunangshu Biswas
  Topic: TBA

Last modified October 29, 2008.