The reflected occupation convolution

Independence at the exponential clock reduces reflection from a level to the convolution of the supremum tail with the exponential drawdown density.

Reference: A. E. Kyprianou, Fluctuations of Lévy Processes with Applications, 2nd ed., Springer, 2014, Chapter 8.

theorem BGW.StableTrees.supOccupationResolvent_eq {θ : ℝ} (hθ : 1 < θ ∧ θ ≤ 2) {q : ℝ} (hq : 0 < q) {ε : ℝ} (hε : 0 < ε) {y : ℝ} (hy : 0 ≤ y) : supOccupationResolvent θ q ε y = ε⁻¹ * ∫ (r : ℝ) in Set.Ioc 0 ε, Real.exp (-(q ^ (1 / θ) * (ε - r))) * supPotential θ q (y - r)

Explicit convolution formula for the occupation resolvent at nonnegative starts.