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

Abstract

We develop a fixed-reference weighted $L^2$ theory for terminal-value fully nonlinear parabolic equations. A prescribed uniformly elliptic diffusion starting from a point determines both the linear reference operator and a space--time occupation measure; time weights of order $α$ accommodate the point-start singularity. For the associated linear backward equation, we identify the optimal source-to-intrinsic-Hessian norm $C_w(X)$ and derive bounds on it from a moving-measure Bochner identity. We compute the sharp Brownian benchmark $C_α^{\mathrm{Br}}=\sup_{n\in\mathbb{N}, n\geq 2}2\sqrt{n(n-1)}/(n-1+α)$, prove that it is a universal tangent lower bound, and show that it is attained by regular weights for time-inhomogeneous affine Gaussian diffusions and is the infimum over such weights for normalized state-dependent covariances. Under the standing data and lower-order assumptions, the Hessian estimate yields our main nonlinear result: if the driver's Hessian Lipschitz constant satisfies $C_w(X)L_H<1$, then the equation has a unique weighted Sobolev solution and the Picard iteration on the source converges geometrically, with no smallness condition on the value or gradient channels. For the Brownian model at order $1/2$, counterexamples in dimensions $d\geq7$ show that this strict threshold cannot be uniformly improved within the occupation-Sobolev class. For weights of order below one, additional spatial regularity and unweighted integrability of the data yield an unweighted spatial jet and a unique solution in the regular fixed-reference Markovian class for second-order backward stochastic differential equations (2BSDEs) defined here.

Disclosure

“f Lemma 7.3 and the coefficient-norm and threshold comparisons used in Theorem 7.4. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the preparation of this work, the authors used OpenAI GPT-5.6 Sol (model identifier gpt-5.6-sol) in order to assist in developing the counterexample constructions and exact matching calculations presented in Theorem 7.1, Lemma 7.3, and Theorem 7.4. After using this tool/service, the authors reviewed”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T05:19:52.437872Z

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