# asymptotic behaviour of principal eigenfunctions and Large Deviations

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?