We study the existence–uniqueness of solution (u,λ) to the ergodic Hamilton–Jacobi equation (Formula presented.) and u≥0, where s∈(12,1). We show that the critical λ=λ∗, defined as the infimum of all λ attaining a non-negative supersolution, attains a nonnegative solution u. Under suitable conditions, it is also shown that λ∗ is the supremum of all λ for which a non-positive subsolution is possible. Moreover, uniqueness of the solution u, corresponding to λ∗, is also established. Furthermore, we provide a probabilistic characterization that determines the uniqueness of the pair (u,λ∗) in the class of all solution pair (u,λ) with u≥0. Our proof technique involves both analytic and probabilistic methods in combination with a new local Lipschitz estimate obtained in this article.
