Twice Fréchet differentiability of the conditioning functional in infinite-dimensional shape bridges

Ascertain whether, for the Hilbert-space representation X_t ∈ L^2(D, R^d) of a shape process induced by a Kunita flow, the conditional expectation E[δ_Γ(\tilde{X}_T) | \tilde{X}_t] is twice Fréchet differentiable when conditioning on X_T ∈ Γ, where Γ ⊂ L^2(D, R^d) has non-zero measure; establish conditions under which this differentiability holds so that the Doob h-transform yields a well-defined stochastic differential equation for the conditioned process.

Background

The authors develop a conditioning framework for nonlinear, infinite-dimensional shape processes using a Doob h-transform in Hilbert spaces. This construction requires the twice Fréchet differentiability of the functional h(x) = E[δ_Γ(\tilde{X}_T) | \tilde{X}_t = x] to derive the drift correction and obtain a well-defined SDE for the conditioned process.

While they provide alternative smooth surrogates (e.g., kernel-based conditioning) that guarantee differentiability, it remains explicitly unknown whether the required differentiability holds when conditioning directly on hitting sets Γ for shape processes, which would enable exact bridge sampling in infinite dimensions.