Ergodic Problems for Viscous Hamilton--Jacobi Equations with Inward Drift

Abstract

In this paper we study the ergodic problem for viscous Hamilton-Jacobi equations with superlinear Hamiltonian and inward drift. We investigate (i) existence and uniqueness of eigenfunctions associated with the generalized principal eigenvalue of the ergodic problem, (ii) relationships with the corresponding stochastic control problem of both finite and infinite time horizon, and (iii) the precise growth exponent of the generalized principal eigenvalue with respect to a perturbation of the potential function.

Sharp estimates of the generalized principal eigenvalue for superlinear viscous Hamilton-Jacobi equations with inward drift

“…See Theorem 3.1 of Section 3 for the explicit form of c 0 . Note that a rough estimate λ max (β) = O(β δm * δm * +η ) has been obtained in [3,Theorem 2.3], but the new ingredient here is that we show existence of a limit of β − δm * δm * +η λ max (β) as β → ∞, together with its explicit form. To our best of knowledge, (1.5) has not been obtained even for the linear operator (1.3), namely, the case where m = 2 in our context.…”

“…In this paper we complete the analysis we performed in the previous paper [3] by providing new results as well as sharp estimates for the generalized principal eigenvalue of superlinear viscous Hamilton-Jacobi equations. More specifically, we consider the following equation with superlinear exponent m > 1 and real parameter β ≥ 0:…”

“…As mentioned, ergodic problem (EP) is a nonlinear extension of linear eigenvalue problem (1.3). From this point of view, a similar criticality theory is discussed in [2,3,7,8] for viscous Hamilton-Jacobi equations of the form (EP) (see also [9] for a discrete analogue of it). The present paper is closely related to [3] where the asymptotic behavior of λ max (β) as β → ∞ is investigated under some conditions including (1.1); it is proved that the spectral function is non-decreasing and concave in [0, ∞) with λ max (0) = 0, and that the limit of λ max (β) as β → ∞ is determined according to the strength of the inward-pointing drift b.…”

“…In view of (1.4), one of the following (a) or (b) happens: (a) there exists a β 0 ∈ (0, ∞) such that λ max (β) = λ max for all β ≥ β 0 ; (b) λ max (β) < λ max for all β ≥ 0. In [3], it is proved that (a) occurs provided (b, V ) satisfies condition (1.1) with a = 0 and 0 < η ≤ 1. In this situation, we face a plateau since λ max (β) does not increase anymore after β 0 .…”

Phase transitions arising in stochastic ergodic control associated with viscous Hamilton–Jacobi equations with bounded inward drift

“…In this appendix, we prove Theorem A.1 below, which is a slight refinement of [12,Théorème IV.1]. This kind of gradient estimates have been used in [3,[7][8][9][10] to obtain a bound of jD/j. Here we establish an estimate which enables one to obtain the decay rate of jD/ðxÞj as jxj !…”