期刊
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
卷 124, 期 -, 页码 169-219出版社
ELSEVIER SCIENCE BV
DOI: 10.1016/j.matpur.2018.05.008
关键词
Generalized principal eigenvalue; Recurrence and transience; Viscous Hamilton-Jacobi equations; Risk-sensitive control; Ergodic control; Semi-linear eigenvalue problems
资金
- Army Research Office [W911NF-17-1-001]
- National Science Foundation [DMS-1715210]
- Office of Naval Research [N00014-16-1-2956]
- INSPIRE faculty fellowship
- DST-SERB [EMR/2016/004810]
This paper studies the eigenvalue problem on R-d for a class of second order, elliptic operators of the form L-f = a(ij) partial derivative(xi) partial derivative(xj) + b(i) partial derivative(xi) + 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. (C) 2018 Elsevier Masson SAS. All rights reserved.
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