On bounded continuous solutions of the archetypal equation with rescaling

The 'archetypal' equation with rescaling is given by y(x)= ∫∫ R² y(a(x b))μ(da, db) (x ∈ R), where μ is a probability measure; equivalently, y(x)= E{y(α(x β))}, with random α, β and E denoting expectation. Examples include (i) functional equation y(x) =Σ_ip_iy(a_i(x b_i)); (ii) functional-differential ('pantograph') equation y(x) + y(x)=Sigma;i piy(ai(x ci)) (pi >0, Σ_ip_i =1). Interpreting solutions y(x) as harmonic functions of the associated Markov chain (X_n), we obtain Liouville-type results asserting that any bounded continuous solution is constant. In particular, in the 'critical' case E{ln |α|}=0 such a theorem holds subject to uniform continuity of y(x); the latter is guaranteed under mild regularity assumptions on β, satisfied e.g. for the pantograph equation (ii). For equation (i) with a_i =q^mi(m_i ∈ ε Z,Σ_i p_im_i =0), the result can be proved without the uniform continuity assumption. The proofs exploit the iterated equation y(x) =E{y(Xτ ) |X₀ =x} (with a suitable stopping time τ ) due to Doob's optional stopping theorem applied to the martingale y(X_n).