A variational formula for risk-sensitive control of diffusions in ${\mathbb{R}}^d$

We address the variational problem for the generalized principal eigenvalue on ${\mathbb{R}}^d$ of linear and semilinear elliptic operators associated with nondegenerate diffusions controlled through the drift. We establish the Collatz-Wielandt formula for potentials that vanish at infinity under minimal hypotheses, and also for general potentials under blanket geometric ergodicity assumptions. We also present associated results having the flavor of a refined maximum principle.