An infinite-dimensional representation of Ray-Knight theorems for the Bass-Burdzy flow

[Bass and Burdzy 1999] introduced a stochastic flow of homeomorphisms of the real line ${\mathcal{R}}_t:x\mapsto {\mathcal{R}}_t(x)$, driven by a Brownian motion. Ray-Knight theorems for the process ${\mathcal{R}}_t^{-1}(0)$ were later obtained by [Hu and Warren 2000]. We extend their results by proving an integral representation driven by a common white noise for the local times of ${\mathcal{R}}_t^{-1}(0)$, in analogy with the case of the standard Ray-Knight theorems [Aïdékon, Hu and Shi 2024]. We generalize this representation to the setting of a Bass-Burdzy flow driven by a drifted Brownian motion perturbed at its maximum, and use this framework to answer a question of [Aïdékon, Hu and Shi 2026] concerning the description of the conditional law of a perturbed reflecting Brownian motion given its occupation measure in terms of Bass-Burdzy flows.