Abstract. 
In these two lectures I plan to introduce the large deviation technique in
stochastic differential equations. In the first lecture, we start from the
standard Brownian motion, construct the action functional, and estimate the
upper/lower bounds of certain "rare events". In lecture II, we use the
large deviation technique to construct action functionals of stochastic
differential equations. Then we will study the first exit time/location
problems and tail of invariant probability measure.

These lectures are based on Random Perturbations of Dynamical Systems by
Yuri Kifer and Random Perturbations of Dynamical Systems by Freidlin and
Wentzell. No specific background in stochastic calculus will be required
for this talk.