Dear Math Overflowers,
I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more details if needed.
I'm working in the 1D domain $\Omega=(0,1)$, I have a fixed weight $\Theta(x)$ which is positive in the interior and vanishes linearly at the boundaries, typically $\Theta(x)=x(1-x)$. Let $\kappa>0$ be a small viscosity parameter, and finally consider a fixed potential $\Phi(x)$ smooth up to the boundary.
It is not too difficult to check that the principal eigenvalue problem $$ \left\{ \begin{array}{ll} -\kappa\partial_{xx}^2 (\Theta u) -\partial_x(\Theta u \partial_x\Phi) =\lambda u & \mbox{in }\Omega \\ \Theta u=0 & \mbox{on }\partial\Omega \end{array} \right. $$ and its adjoint $$ \left\{ \begin{array}{ll} -\kappa\Theta\partial_{xx}^2 v +\Theta \partial_x\Phi \partial_x v =\lambda v & \mbox{in }\Omega \\ v=0 & \mbox{on }\partial\Omega \end{array} \right. $$ are well-posed (note however that $\Theta$ vanishes at the endpoints, so this is perhaps not completely trivial -- but true nonetheless). Moreover, I choose to normalize my eigenfunctions $u,v>0$ in such a way that $$ \int_0^1 u=\int_0^1 uv =1. $$ Emphasizing now the dependence on $\kappa$, let me define the probability measure $$ \pi_\kappa:=u_\kappa(x)v_\kappa(x) dx. $$
Question: is there any standard way to prove that, in the vanishing viscosity limit $\kappa\to 0$, the sequence $(\pi_\kappa)_{\kappa>0}$ satisfies a Large Deviation Principle with speed $\frac 1\kappa$ and rate function $\Phi(x)$ precisely given by my prescribed potential?
I suspect that the Freidlin-Wentzell theory should help, but I am not as acquainted with probability theory as I would like to be... Also, before trying brute force and spending some time trying to get fine information on each eigenfunction $u_\kappa$ and $v_\kappa$ separately (in order to take the product in the end), I am wondering if there might be a lazy way around and only talk of the product itself? At the PDE level both the non self-adjointness and the degeneracy of the diffusion ($\Theta$ vanishes) make the problem tricky. I have no clue about what's going on at the probabilistic level, but this looks like some product of "forward $\times$ backward" Kolmogorov operators/eigenfunctions, so perhaps there is some literature out there?
