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?

Your Answer

 
discard

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.