Rafał Kapica

ON A REFINEMENT TYPE EQUATION

Abstract Let (Ω, A, P) be a complete probability space. We show that the trivial function is the unique L 1-solution of the following refinement type equation for a wide class of the given functions φ. This class contains functions of the form φ(x, ω) = α(ω)x – β(ω) with –∞ < ∫Ω log|α(ω)|dP(ω) < 0.

Refinement equations and distributional fixed points

Abstract Connections between L 1 -solutions of the refinement equation of the form f ( x ) = ∫ Ω | L | f ( Lx - M ) dP and distributional fixed points of a special random affine map are investigated. Using results on convergence of perpetuities the direct form of the Fourier transform of L 1 -solutions of the above refinement equation is obtained.

Random iteration and Markov operators

Assume that is a probability space and is a metric space. Given a product measurable function , we examine connections between the iterates (in the sense of K. Baron and M. Kuczma, Colloq. Math. 37 (1977), 263–269) of and the Markov operator with adjoint of the form . Moreover, some results concerning the existence and the uniqueness of solutions of the equation will be also presented.

Inhomogeneous refinement equations with random affine maps

Given a probability space , random variables , and a function , we obtain two characterizations of these , which are solutions of the inhomogeneous refinement equation with a random affine map of the form .

Matrix refinement type equations

Abstract Taking advantage of perpetuities and the asymptotic behavior of products of random matrices we obtain the direct form of the Fourier transform of an L 1 -solution of the following random matrix refinement type equation f ( x ) = ∫ Ω | det L ( ω ) | C ( ω ) f ( L ( ω ) x - M ( ω ) ) P ( d ω ) , where M : Ω → R p is a random vector and L : Ω → R p × p , C : Ω → R q × q are random matrices. Moreover, the dimension of the space of all L 1 -solutions of this refinement type equation is examined.

A uniqueness-type problem for linear iterative equations

Abstract Given a probability space (Ω,A,P), a Polish space X and a (B(X) ⊗ A)-measurable function f:X × Ω→X we consider Borel and bounded solutions ψ:X→ℝ of the functional inequality ψ(x) ≤ ∫Ωψ(f(x,ω))P(dω) and a uniqueness-type problem for Borel and bounded solutions φ:X→ℝ of the functional equation φ(x) = ∫Ωφ(f(x,ω))P(dω).