Malliavin Calculus of Wiener Chaos Expansions on Fluctuating Event Hypersurfaces

Mathematical Applications of Science Fiction We construct a rigorous framework for the Malliavin calculus ofvariations applied to functionals defined on stochastically fluc-tuating event hypersurfaces Σ𝑡 (𝜔) embedded in a Riemannianmanifold 𝑀. By lifting the geometric flow to the frame bundleof the Wiener space W, we derive a generalized Clark-Oconeformula for the chaos decomposition of boundary-restrictedobservables. We introduce the Thorne-Watanabe derivative, acovariant operator acting on sections of the derived bundle ofHida distributions, and prove its commutativity with the spectralprojection operators of the Ornstein-Uhlenbeck semigroup, mod-ulo a curvature correction term involving the second fundamentalform of Σ𝑡 . The analysis culminates in a stability theorem for the𝐿2-convergence of Wiener chaos expansions under perturbationsof the underlying motivic measure, with applications to non-perturbative quantum gravity and holographic renormalizationon rough boundaries.