The ‘archetypal’ equation with rescaling is given by (), where μ is a probability measure; equivalently, , with random α,β and denoting expectation. Examples include (i) functional equation ; (ii) functional–differential (‘pantograph’) equation (pi>0, ). Interpreting solutions y(x) as harmonic functions of the associated Markov chain (Xn), we obtain Liouville-type results asserting that any bounded continuous solution is constant. In particular, in the ‘critical’ case 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 ai=qmi (, ), the result can be proved without the uniform continuity assumption. The proofs exploit the iterated equation (with a suitable stopping time τ) due to Doob's optional stopping theorem applied to the martingale y(Xn).
- Received May 28, 2015.
- Accepted June 25, 2015.
- © 2015 The Author(s)
Published by the Royal Society. All rights reserved.