We study weighted $L^2$ Hessian estimates under diffusion marginals, allowing the equation's principal matrix to be chosen independently. For regular uniformly elliptic coefficients, bounded reference drift and compactly supported doubling initial laws, the optimal large-damping constant on the full interval is the maximum of initial, flat and propagation contributions. Finiteness yields weighted Sobolev well-posedness and a contraction criterion for nonlinear perturbations. An explicit operator formula determines the initial contribution; propagation bounds become exact under uniform directional limits or asymptotic isotropy at infinity. Under geometric assumptions including nonnegative Ricci curvature, propagation is determined by terminal momenta of action-minimizing paths. For reference covariance $a$, matching principal matrix $a/2$ and Hessian normalization $a^{1/2}D^2u\,a^{1/2}$, every initial law gives a finite estimate. Without an additional time weight, the limiting constant lies between $2$ and $2\sqrt2$: finite atomic laws attain the upper bound, while nondegenerate Gaussian laws and their finite mixtures attain the lower bound.
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 $\alpha$ 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 $C_\alpha^{\mathrm{Br}}=\sup_{n\in\mathbb{N}, n\geq 2}2\sqrt{n(n-1)}/(n-1+\alpha)$, 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.
Jihao Long, Zhenhu Zhao· 0 citations
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