I will discuss a number of examples where the addition of noise leads to a qualitative change in the dynamics of the system. My main emphasis will be on a simple planner ODE which has solutions which explode in finite time. Yet when noise is added the system is stabilized and develops a number of interesting "out of equilibrium" behavior. This is an istructive example of "stablisation by noise" and could be an instructive example to consider morcomplicated settings such as PDEs which people hope are stabilised by noise. In this simple example, the interaction of the noise and the instability leads to a system which at once equilibrates quickly but also has heavy tails and "intermittent" behavior. I will provide a general framework to think about such problems.

If time permits I will also talk about a simple example which deterministically possess many invariant measure, yet the zero noise limit of the stochastically forced system limit posseses only one invariant measure. The problems posseses some serious mathematical chalaenges including a limiting problem which does not have unique solutions. Uniquness is only obtained when it is views as the limit of a noisy system.