I will describe a class of examples which are deterministically unstable but in which the addition of any noise leads to global stability. The analysis will consist of constructing an Lyapunov function adapted to the dynamics by careful matched asymptotics. I will show how the constructed function gives sharp bounds on the decay of the invariant measure which surprisingly will be polynomial despite the addition of uniformly elliptic additive noise. Furthermore despite the heavy tailed invariant measure, I will show that the system mixed extremely fast. Our method is fairly algorithmic and hopefully can be applied to many problems. 

Joint work with David Herzog.