# Student Probability Seminar

#### Laplace's Method and Exit Problems

Speaker: Hong-Bin Chen, CIMS

Location: Warren Weaver Hall 1314

Date: Monday, March 11, 2019, 12:30 p.m.

Synopsis:

In the study of random perturbations of dynamical systems, Freidlin and Wentzell introduced the notion of action function(al)s, which is a framework analogous to Laplace's method in real analysis. Many interesting results were obtained by this framework. In particular, consider the equation ẋ=b(x) with a stable equilibrium at 0. Let D be a suitable domain containing 0. Then deterministically, the flow starting from 0 cannot escape D. However, after adding noise in the form of εdW to the equation, there is a positive probability for this diffusion process originated from 0 to exit D eventually. The asymptotic location of the exit point(s) on the boundary as ε→0 is of interest. We will show a theorem of this type using action functionals.