Random Frame Decompositions from Weighted Residual Flows

Abstract: We study the evolution of a positive operator under weighted residual maps determined by a finite family of orthogonal projections. Iterating these maps along the rooted tree of multi-indices produces a "weighted residual energy tree", together with natural path measures obtained by normalizing the dissipated energy or trace at each step. Under a quantitative coverage condition on the projections, we show that along almost every branch the residuals converge strongly to zero and the dissipated pieces admit a rank-one decomposition that reconstructs the initial operator. In the special case where the initial operator is the identity on a subspace, this yields almost surely a random Parseval frame generated intrinsically by the weighted residual dynamics.