On probabilistic approach to the eigenvalue problem for maximal elliptic operator (Q1567130)

Let \(D\) be a bounded domain in \(\mathbb{R}^d\) with smooth boundary \(\partial D\). Let \(L^\alpha\), \(\alpha\in A\) (\(A\) is a given set), be nondegenerate second-order linear differential operators with parameter \(\alpha\), let also \(L= \sup_{\alpha\in A}L^\alpha\). We consider the Dirichlet eigenvalue problem with respect to \(L\) on \(D\): \[ Lu+\lambda u= 0\quad\text{on }{\mathcal D},\quad u>0\quad\text{on }{\mathcal D},\quad u/\partial{\mathcal D}= 0.\tag{1} \] It is supposed that there exist an eigenvalue \(\lambda\) and a corresponding (smooth) eigenfunction \(u\) satisfying (1). Various properties of \(\lambda\) are discussed. An probabilistic representation for \(\lambda\) is obtained.