I'm almost not at all knowledgable in either Freidlin-Wentzel theory or Kramers' escape problem as it is known in the physics community, so please excuse some of my naivety.


One can use Freidlin-Wentzel theory to study the problem of a particle with position given by the SDE


$$dX\_t = V(X\_t)\,dt + \sqrt{\epsilon}\,dW\_t (\*)$$


escaping from a local minimum over some barrier, or more precisely derive the mean first passage time over that barrier. To this end one has to introduce the concept of large deviation principles, prove Schilder's theorem, then transfer the LDP for the Brownian motion $W$ to $X$ via the Ito map. Then it seems one can *start* studying the escape problem.


What suprises me that in the physics/chemistry/etc. community they study the same (?) problem, but their derivation of the mean passage time makes no reference at all to Large deviations and proceeds more or less elementary using the Fokker-Planck equation associated with (\*) (cf. e.g. [these lectures notes](http://physics.gu.se/%7Efrtbm/joomla/media/mydocs/LennartSjogren/kap8.pdf)).


How do these two approaches relate to each other? The physics approachs seems so much easier that it makes me question the merit of studying Freidlin-Wentzel theory to solve escape problems.



Kramers' escape problem: statistical physics vs. Large deviations
The Kramers theory is limited to reversible processes in equilibrium, while the Freidlin-Wentzel theory generalises this to irreversible processes out of equilibrium.


The distinction appears in the stochastic differential equation
$$dx\_t=f(x\_t)dt+ \sqrt{2\epsilon}dW\_t,$$
in the Kramers theory it is assumed that $f(x)=-\nabla U$ is the gradient of a potential $U:\mathbb{R}^d\rightarrow\mathbb{R}$, and the escape rate is given in terms of this potential. The system is in equilibrium with probability distribution $p(x)\propto e^{-U/\epsilon}$. The Freidlin-Wentzel theory applies to more general functions $f$, and can describe systems out of equilibrium


