Vanishing discount problem and the additive eigenvalues on changing domains

Abstract

We study the asymptotic behavior, as λ → 0 + , of the state-constraint Hamilton-Jacobi equationHere, Ω is a bounded domain of R n and φ(λ), r(λ) : (0, ∞) → (0, ∞) are continuous nondecreasing functions such that lim λ→0 + φ(λ) = lim λ→0 + r(λ) = 0. A similar problem on (1 + r(λ))Ω is also considered. Surprisingly, we are able to obtain both convergence results and non-convergence results in this setting. Moreover, we provide a very first result on the asymptotic expansion of the additive eigenvalue of H in (1 ± r(λ))Ω as λ → 0 + . I ntroductionLet φ(λ), r(λ) : (0, ∞) → (0, ∞) be continuous nondecreasing functions such that lim λ→0 + φ(λ) = lim λ→0 + r(λ) = 0. We study the asymptotic behavior, as the discount factor φ(λ) goes to 0, of the viscosity solutions to the following state-constraint Hamilton-Jacobi equationHere, Ω is a bounded domain of R n . For simplicity, we will write Ω λ = (1 − r(λ))Ω and Ω λ = (1 + r(λ))Ω for λ > 0. A similar problem for Ω λ will also be considered. Roughly speaking, along some subsequence λ j → 0 + , we obtain the limiting equation as a state-constraint ergodic problem: H(x, Du(x)) c 0 in Ω, H(x, Du(x)) c 0 on Ω.

Selection problems in Large Deviations in Games under the logit choice protocol

“…It is worth noting that (TP) keeps all of its characteristics and properties in X r . For state-constraint Hamilton-Jacobi equations in nested domains, see [5,2,6,15].…”