# Student Probability Seminar

#### Rare exit events near a repelling equilibrium

**Speaker:**
Hong-Bin Chen, CIMS

**Location:**
Warren Weaver Hall 201

**Date:**
Thursday, October 24, 2019, 12:15 p.m.

**Synopsis:**

Consider a deterministic ODE αΊ=b(x) perturbed by white noise with amplitude ε, which results in an SDE. Let D be a domain containing 0 and suppose that, in D, the vector field b has only one equilibrium, which is assumed to be at 0. Let π be the time for X_t, the solution to the SDE, to exit D. When the equilibrium is stable, the exit location X_π has been studied using Freidlin and Wentzell large deviation theory. In particular, rare exit events exhibit an exponential decay in ε as ε→0. However, when the equilibrium is unstable, the decay rate of rare exit events is expected to be polynomial in ε, which is out of the reach of FW theory. In this talk, we will first show a few simple computations in 2D or 3D to illustrate this phenomenon. Then, the general scenario in any finite dimension and an outline of arguments will be described.