Strict monotonicity of principal eigenvalues of elliptic operators in Rd and risk-sensitive control

Abstract

This paper studies the eigenvalue problem on R d for a class of second order, elliptic operators of the form L f = a ij ∂ xi ∂ xj + b i ∂ 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.

Certain Liouville properties of eigenfunctions of elliptic operators

“…We also assume that λ * (V ) is finite. Let us begin with the following equivalence between the strict right monotonicity of the principal eigenvalue and the criticality of the operator [5]. See also [40,Theorems 4.3.3 and 7.3.6] for similar results concerning operators with regular coefficients.…”

A Variational Formula for Risk-Sensitive Control of Diffusions in $\mathbb{R}^d$

“…As noted in [3], Assumption 2.9 does not hold for diffusions with bounded a, and b. Therefore, to treat this case, we consider an alternate set of conditions.…”

“…Therefore, to treat this case, we consider an alternate set of conditions. The eigenvalue λ * (G) in (2.10) represents the optimal risk-sensitive ergodic cost [1,3,10,11]. In order to define this control problem, we need to introduce some additional notation.…”

“…The motivation for this work is the infinite horizon risk-sensitive control problem on the entire domain, which seeks to minimize the asymptotic growth rate of the expected 'exponential of integral' cost, and which, under suitable assumptions, coincides with the generalized principal eigenvalue of the associated semilinear elliptic operator (for some recent results see [1,3]). Recall the celebrated formula of Donsker-Varadhan: for a uniformly elliptic nondivergence form operator L on a smooth bounded domain D ⊂ R d , the principal eigenvalue λ 1 (L, D) can be expressed as λ 1 (L, D) = sup ϕ∈C 2,+ (D) domains.…”

“…For a continuous function g : R d → (0, ∞) which is bounded below away from 0, we let O(g) denote the space of continuous functions f : R d → R satisfying sup x∈R d |f (x)| g(x) < ∞, and by o(g) is subset consisting of those functions which satisfy lim sup R→∞ sup x∈B c R |f (x)| g(x) = 0.We borrow the following result from[3].…”