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

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Abstract
Conditions are derived under which a probability measure on the Borel subsets of [0, ∞) is a stationary distribution for the content {Xt} of an infinite dam whose cumulative input {At} is a pure-jump Lévy process and whose release rate is a non-decreasing continuous functionr(·) of the content. The conditions are used to find stationary distributions in a number of special cases, in particular whenand whenr(x) =xαand {At} is stable with index β ∊ (0, 1). In general ifEAt, < ∞ andr(0 +) > 0 it is shown that the condition supr(x)>EA1is necessary and sufficient for a stationary distribution to exist, a stationary distribution being found explicitly when the conditions are satisfied. If supr(x)>EA1it is shown that there is at most one stationary distribution and that if there is one then it is the limiting distribution of {Xt} ast→ ∞. For {At} stable with index β andr(x) =xα, α + β = 1, we show also thatcomplementing results of Brockwell and Chung for the zero-set of {Xt} 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.

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