  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.
