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

- Series
- Other Talks
- Time
- Friday, November 6, 2009 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 154
- Speaker
- Sergio Almada – 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.