On bounded continuous solutions of the archetypal equation with rescaling

Abstract

The ‘archetypal’ equation with rescaling is given by y ( x ) = ∬ R 2 y ( a ( x − b ) ) μ ( d a , d b ) ( 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 ) = ∑ i p i y ( a i ( x − b i ) ) ; (ii) functional–differential (‘pantograph’) equation y ′ ( x ) + y ( x ) = ∑ i p i y ( a i ( x − c i ) ) ( p i >0, ∑ i p 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 m i ( m i ∈ Z , ∑ i p i m i = 0 ), the result can be proved without the uniform continuity assumption. The proofs exploit the iterated equation y ( x ) = E { y ( X τ ) | X 0 = x } (with a suitable stopping time τ ) due to Doob's optional stopping theorem applied to the martingale y ( X n ).