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Exercise 1. Three balls are randomly chosen with replacement from an urn containing 5 blue, 4 red, and 2 yellow balls. Let X denote the number of red balls chosen. (a) What are the possible values of X? (b) What … Continue reading
Practice midterm 1 We will do the correction in class on 09/28.
In the previous lecture we defined the Young’s integral when and with . The integral path has then a bounded -variation. Now, if is a Lipschitz map, then the integral, is only defined when , that is for . With … Continue reading
Exercise 1. Solve Exercise 44 in Chapter 1 of the book. Exercise 2. Solve Exercise 3 in Chapter 1 of the book. Exercise 3. Solve Exercise 39 in Chapter 1 of the book. Exercise 4. The heat kernel on is given by . By … Continue reading
Exercise 1. The Hermite polynomial of order is defined as Compute . Show that if is a Brownian motion, then the process is a martingale. Show that Exercise 2. (Probabilistic proof of Liouville theorem) By using martingale methods, prove that if … Continue reading
Solve Problems 1,2,8,9,10,11 in Milnor’s book. (Extra credit for problem 6)