Safe Control in the Presence of Stochastic Uncertainties

The paper’s Theorem 1 states that the CCDF F(z,T;ℓ)=P(Φ_x(T)≥ℓ) for the augmented diffusion Z_t=[φ(X_t);X_t] solves the initial–boundary value problem ∂_T F = 1/2 ∇·(D∇F) + L_ρ F − 1/2 ∇·D F on {z: z[1]≥ℓ} with Dirichlet boundary F=0 on {z[1]<ℓ} and initial condition F(z,0;ℓ)=1_{z[1]≥ℓ}, where D=ζζ^⊤ (eq. (25.B)) . The proof sketched in the paper uses a Feynman–Kac construction with a killing rate γ1_{M^c} and then sends γ→∞ to encode absorption outside M, and finally rewrites the operator via the identity ∇·(D∇F)=Tr(D Hess F)+(∇·D)·∇F (their eq. (42) with footnote 19) . The candidate solution reaches the same PDE and boundary/initial conditions by a standard “killed-process” backward equation/Dynkin’s formula argument on the augmented process, also invoking the same divergence identity. Thus, the results agree and the proofs are genuinely different in technique (Feynman–Kac limit vs. killed-generator/Dynkin). Minor notational slips in the paper (e.g., writing Lρ without an explicit F in (25.B), and the shorthand in (42)) are clarified by context and footnote 19 . The paper’s assumptions (unique solution for the SDE; φ∈C^2 with nonvanishing gradient on ∂C) are stated earlier and are consistent with what is needed for both approaches . The only gap is that the γ→∞ limit and boundary regularity are not fully justified in print, but these are standard and can be completed with routine arguments. Overall: both correct; different proofs.