| In Chapter 1, I derive a version of the central limit theorem for an Isotropic Brownian Flow (IBF) fst (·) on Rn , n ≥ 2, with a positive Lyapunov exponent: given any two distinct points x and y in Rn , then the joint distribution f0tx t,f 0ty t converges to a multivariate normal distribution when t approaches infinity.; In Chapter 2, I first show that under the action of an IBF with positive Lyapunov exponent, the probability of a non-trivial set gamma0 always stays away from a fixed ball BR( A), and decays at a faster than any polynomial rate. Then I discover that within a short time interval, a small bounded set has a strictly positive probability to follow a constant flow. Using this result, I show the set of points visited by a non-trivial set, called the contaminated set, expands at a linear speed.; In Chapter 3, I study the backward stochastic differential equations in higher dimensions with cylindrical reflecting boundary (BSDERs). I apply the penalization method, and prove the convergence of the penalization process, which established an existence result for adapted solutions of such equations. This result is a generalization of the work of Cvitanic and Karatzas. |
