Probabilistic approach for comparing first eigenvalues (Q1058235)

A probabilistic method is used to compare the first eigenvalues of the problems \[ G\psi +\lambda \psi =0,\quad G\psi -q\psi +\lambda \psi =0, \] \[ G\psi -q\psi +\lambda \rho \psi =0\quad in\quad D,\quad \psi |_{\partial D=0}, \] when G is the generator of a certain class of Hunt processes \((X_ t)\). By considering subordinated and time changed processes obtained from the process \((X_ t)\) and characterizing first eigenvalues as \(\lambda_ 1=\sup \{\lambda:\sup_{x\in D}E^ x[e^{\lambda T}]<\infty \}\), the comparison of first eigenvalues can be reduced to the comparison of stopping times. The results generalize classical ones for the Laplace operator. The paper also contains a section with various examples.