Fully nonlinear parabolic equations under a fixed reference diffusion:weighted $L^2$ Hessian estimates and well-posedness

What the math gives to ML

The paper provides a constructive contraction principle for fully nonlinear parabolic equations: measure curvature using the diffusion-intrinsic Hessian σD²uσ and evaluate errors under the reference diffusion's occupation measure with a time-singular weight. The explicit condition C_w(X)L_H<1 yields geometric convergence of source Picard iteration without requiring smallness in value or gradient channels. This can be transferred to PINNs and neural PDE solvers by using diffusion-based collocation, intrinsic Hessian residuals, and an online estimate of the nonlinear driver's curvature Lipschitz constant. The Brownian benchmark supplies a computable stability threshold and an ablation target.