Brownian Motion (Fall 2009)
Manjunath Krishnapur
Tuesday, Thursday 2:30 - 4:00 (PM), in LH III
Plan of the course (tentative)
- Overview. [First meeting on 4th August]
- Definition and Levy's construction C[0,1], C[0,∞), their Borel sigma fields, d-dimensional BM.
- Continuity properties - Modulus of continuity. Hölder 1/2-ε.
- Nowhere differentiable. Nowhere Hölder 1/2+ε
- Gaussian process, covariance. Invariance Time reversal, Scaling, Rotations.
- Markov processes. Transition probability. Markov property of BM. Heat equation.
- Filtrations, Stopping times. Zero-one laws (Blumenthal, Kolmogorov).
- Strong Markov property of BM.
- Applications: LLN, Zero set is perfect.
- Reflection principle. P(M_{t}>x)=2P(B_{t}>x). First passage times - P(T_{a}≤ t)=P(M_{t}>a).
- M_{t}-B_{t}\eqd |B_{t}|. Remark about M_{t} as local time.
- Fractals Minkowski and Hausdorff dimensions. Deterministic examples, upper bounds.
- Energy criterion. Lower bounds for Hausdorff dimensions.
- First, Second and modified first moment methods.
- Dimension of random subsets of [0,1], a general lemma (using `correlations') .
- Random fractals: Graph, Path, Record times, Zero set,
- Martingales Recap of discrete martingales. Optional stopping. Convergence theorem.
- Application to absolute continuity question. Kakutani's criterion.
- B+f are mutually abs. cts. to B. What is the Radon-Nikdym derivative?
- Continuous martingales. B_{t}, (B_{t})^{2}-t, Exponential martingale. Higher dimensions.
- Application to expected hitting time. Exit time of an interval.
- Discrete and continuous Random walks.
- Skorokhod's embedding theorem.
- Donsker's theorem. Applications: Arcsine laws. Maxima etc.
- LIL (from BM to RW). Mention functional LIL.
- No points of increase (from RW to BM). But lots of local maxima and minima.
- Random walks and discrete harmonic functions. BM and harmonic functions.
- Recurrence and transience of BM.
- Which sets are hit? Polar sets and capacity for Markov chains and BM
- Dirichlet problem Using BM to solve Dirchlet problem.
- Occupation measure and Green's functions. Harmonic measure.
- Stochastic Calculus Ito's formula. Levy's criterion. More martingales from BM.
- Conformal invariance Winding number. Proof of Picard's theorem. Viràg's lemma.
- Story of Restriction measures and self-avoiding walk.
- Series expansion of BM Eigenfunctions of the Laplacian. Gaussian free field.