Inverse problems for random differential equations using the collage method for random contraction mappings

In this paper we are concerned with differential equations with random coefficients which will be considered as random fixed point equations of the form T(ω,x(ω))=x(ω)T(ω,x(ω))=x(ω), ω∈Ωω∈Ω. Here T:Ω×X→XT:Ω×X→X is a random integral operator, (Ω,F,P)(Ω,F,P) is a probability space and XX is a complete metric space. We consider the following inverse problem for such equations: Given a set of realizations of the fixed point of TT (possibly the interpolations of different observational data sets), determine the operator TT or the mean value of its random components, as appropriate. We solve the inverse problem for this class of equations by using the collage theorem for contraction mappings.