# Strict monotonicity of principal eigenvalues of elliptic operators in $\mathbb{R}^d$ and risk-sensitive control

## TL;DR

This paper investigates the principal eigenvalues of elliptic operators in ^d, linking their strict monotonicity to ergodic properties and applying these insights to risk-sensitive control problems for diffusions.

## Contribution

It characterizes the strict monotonicity of principal eigenvalues in relation to ergodic properties and extends results to equations with measurable coefficients, also establishing duality in ergodic control.

## Key findings

- Strict monotonicity characterizes ergodic properties and uniqueness of ground states.
- Established strong duality for ergodic control linear programming formulations.
- Proved existence and optimality of Markov controls in risk-sensitive control problems.

## Abstract

This paper studies the eigenvalue problem on $\mathbb{R}^d$ for a class of second order, elliptic operators of the form $\mathscr{L} = a^{ij}\partial_{x_i}\partial_{x_j} + b^{i}\partial_{x_i} + f$, associated with non-degenerate diffusions. We show that strict monotonicity of the principal eigenvalue of the operator with respect to the potential function $f$ fully characterizes the ergodic properties of the associated ground state diffusion, and the unicity of the ground state, and we present a comprehensive study of the eigenvalue problem from this point of view. This allows us to extend or strengthen various results in the literature for a class of viscous Hamilton-Jacobi equations of ergodic type with smooth coefficients to equations with measurable drift and potential. In addition, we establish the strong duality for the equivalent infinite dimensional linear programming formulation of these ergodic control problems. We also apply these results to the study of the infinite horizon risk-sensitive control problem for diffusions, and establish existence of optimal Markov controls, verification of optimality results, and the continuity of the controlled principal eigenvalue with respect to stationary Markov controls.