A VARIATIONAL FORMULA FOR RISK-SENSITIVE CONTROL OF DIFFUSIONS IN Rd

We address the variational problem for the generalized principal eigenvalue on 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.