An application of functional equations for generating ε-invariant measures

2018/10/10 by Morawiec, Janusz, Zürcher, Thomas
#39B12 #47A50 #47B38 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 37A30 #Secondary 28A80

paper · doi:10.48550/arxiv.1810.04530

Abstract

Let (X,\mathcal A,μ) be a probability space and let S\colon X→ X be a measurable transformation. Motivated by the paper of K. Nikodem [Czechoslovak Math. J. 41(116) (4) (1991) 565--569], we concentrate on a functional equation generating measures that are absolutely continuous with respect to μ and ε-invariant under S. As a consequence of the investigation, we obtain a result on the existence and uniqueness of solutions φ∈ L¹([0,1]) of the functional equation φ(x)=∑_n=1^N|f_n'(x)|φ(f_n(x))+g(x), where g∈ L¹([0,1]) and f₁,…,f_N\colon[0,1]→[0,1] are functions satisfying some extra conditions.

Related

• Some Class of Linear Operators Involved in Functional Equations
• Existence of invariant probability measures for functional McKean-Vlasov SDEs
• Invariant measures for stochastic functional differential equations
• Measurable solutions of an alternative functional equation
• Invariant Measure for Stochastic Functional Differential Equations in Hilbert Spaces