Small noise limit for dynamics near unstable critical points (Oral Comprehensive Exam).

Other Talks
Friday, November 6, 2009 - 15:00
1 hour (actually 50 minutes)
Skiles 154
Georgia Tech
We consider the Stochastic Differential Equation $dX_\epsilon=b(X_\epsilon)dt + \epsilon dW$ . Given a domain D, we study how the exit time and the distribution of the process at the time it exits D behave as \epsilon goes to 0. In particular, we cover the case in which the unperturbed system $\frac{d}{dt}x=b(x)$ has a unique fixed point of the hyperbolic type. We will illustrate how the behavior of the system is in the linear case. We will remark how our results give improvements to the study of systems admitting heteroclinic or homoclinic connections.  We will outline the general proof in two dimensions that requires normal form theory from differential equations. For higher dimensions, we introduce a new kind of non-smooth stochastic calculus.