Stationary distributions for dams with additive input and content-dependent release rate

Abstract

Conditions are derived under which a probability measure on the Borel subsets of [0, ∞) is a stationary distribution for the content {X_t} of an infinite dam whose cumulative input {A_t} is a pure-jump Lévy process and whose release rate is a non-decreasing continuous function r(·) of the content. The conditions are used to find stationary distributions in a number of special cases, in particular when and when r(x) = x^α and {A_t} is stable with index β ∊ (0, 1). In general if EA_t, < ∞ and r(0 +) > 0 it is shown that the condition sup r(x)>EA₁ is necessary and sufficient for a stationary distribution to exist, a stationary distribution being found explicitly when the conditions are satisfied. If sup r(x)>EA₁ it is shown that there is at most one stationary distribution and that if there is one then it is the limiting distribution of {X_t} as t → ∞. For {A_t} stable with index β and r(x) = x^α, α + β = 1, we show also that complementing results of Brockwell and Chung for the zero-set of {X_t} in the cases α + β < 1 and α + β > 1. We conclude with a brief treatment of the finite dam, regarded as a limiting case of infinite dams with suitably chosen release functions.