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

TL;DR

This paper develops a variational formula for the principal eigenvalue of controlled diffusion operators in , providing new theoretical tools for risk-sensitive control analysis under minimal and ergodic conditions.

Contribution

It introduces a Collatz-Wielandt formula for the principal eigenvalue of controlled diffusions, extending existing results to broader potential functions and ergodic settings.

Findings

Established Collatz-Wielandt formula for potentials vanishing at infinity.

Extended the formula to general potentials under geometric ergodicity.

Provided results akin to a refined maximum principle.

Abstract

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.