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

Abstract

This paper studies the eigenvalue problem on ${\mathbb{R}}^d$ for a class of second order, elliptic operators of the form $\mathcal{L}=a^{ij}{\partial }_{x_i}{\partial }_{x_j}+b^i{\partial }_{x_i}+f$, 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.

45 pages

References in corpus (2)

• On the structure of solutions of ergodic type Bellman equation related to risk-sensitive control
• Zero-sum stochastic differential game with risk-sensitive cost

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• On the policy improvement algorithm for ergodic risk-sensitive control
• Nonzero-sum risk-sensitive continuous-time stochastic games with ergodic costs
• Generalized principal eigenvalues of convex nonlinear elliptic operators in ${\mathbb{R}}^N$
• Generalized principal eigenvalues on ${\mathbb{R}}^d$ of second order elliptic operators with rough nonlocal kernels
• Asymptotics of impulse control problem with multiplicative reward