A Note on the Interpretation of Distributed Delay Equations

The paper explicitly distinguishes sample-averaging of solutions to a random-delay DDE from the distributed-delay DDE that averages the right-hand-side operator, deriving for the discrete case v_R'(t) = α∑_i p_i v_i(t−Δ_i) and v_D'(t) = α∑_i p_i v_D(t−Δ_i), and emphasizing that these are different objects (averaging solutions vs averaging operators) . The candidate solution reproduces the same logic, but adds standard well-posedness and regularity assumptions (method of steps; absolute continuity; SLLN) and explicitly notes when the two coincide only in special cases. Hence, both are correct with essentially the same proof structure.