Stochastic modification of Newtonian dynamics and Induced potential - application to spiral galaxies and the dark potential

The paper proves the stochastic Hamilton–Jacobi equation for the complex action A by inserting the gradient-velocity representation D_μX_t = (1/m)∇A into the differential stochastic Newton equation and applying the chain rule for D_μ, obtaining ∇[∂_tA + (1/(2m))|∇A|^2 + iμ(σ^2/2)ΔA] = −∇U and hence ∂_tA + (1/(2m))|∇A|^2 + iμ(σ^2/2)ΔA = −U (Theorem 6) . The candidate solution follows the same steps, explicitly computing the transport term (∇A·∇)∇A = (1/2)∇|∇A|^2/m and noting the possible additive function of time c(t), then fixing the time-gauge—an innocuous refinement not made explicit in the paper. The action representation used (V = ∇A/m) is exactly Lemma 5 in the paper , and the chain-rule step mirrors equation (76) in the proof of Theorem 6 . Overall, both arguments are correct and essentially identical; the model adds a minor clarification (time-gauge).