'American Institute of Mathematical Sciences (AIMS)'
Publication date
15/01/2015
Field of study
We study the archetypal functional equation of the form y(x)=∬R2y(a(x−b))μ(da,db) (x∈R), where μ is a probability measure on R2; equivalently, y(x)=E{y(α(x−β))}, where E is expectation with respect to the distribution μ of random coefficients (α,β). Existence of non-trivial (i.e. non-constant) bounded continuous solutions is governed by the value K:=∬R2ln∣a∣μ(da,db)=E{ln∣α∣}; namely, under mild technical conditions no such solutions exist whenever K0 (and α>0) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with (α,β). Further results are obtained in the supercritical case K>0, including existence, uniqueness and a maximum principle. The case with P(α0 is drastically different from that with α>0; in particular, we prove that a bounded solution y(⋅) possessing limits at ±∞ must be constant. The proofs employ martingale techniques applied to the martingale y(Xn), where (Xn) is an associated Markov chain with jumps of the form x⇝α(x−β)