Optimal Weighted $L^2$ Hessian Estimates for Parabolic Equations under General Diffusion Marginals

Abstract

We study weighted $L^2$ Hessian estimates for parabolic terminal equations under the marginals of a prescribed diffusion, allowing singular initial laws and different principal coefficients in the equation and the diffusion. Under coefficient regularity, uniform ellipticity and bounded reference drift, we characterize the limiting optimal constant uniformly over shrinking subintervals with zero terminal data. Its finiteness is equivalent to linear well-posedness in the weighted parabolic Sobolev space on the full interval. For initial laws satisfying weighted doubling and coercivity conditions, this constant is the maximum of contributions from heat evolutions of rescaled initial measures, constant weights and exponential mixtures. The phase contributions are bounded above in terms of the coefficients at time zero outside the initial support and at spatial infinity. Under radial reference structure, we obtain exact coefficient formulas invariant under bounded changes of reference drift. In the matching case, with reference covariance $a$, principal matrix $a/2$ and normalized Hessian $a^{1/2}D^2u\,a^{1/2}$, the constant is $2$ for every initial law with time weight $t^\alpha$, $\alpha\ge1/2$; point starts attain $2\sqrt2$ at $\alpha=0$. Whenever the limiting constant is finite, increasing damping makes the full-interval Hessian constant converge to it and the value and gradient gains tend to zero. As applications, we obtain nonlinear solvability and residual stability for small Hessian perturbations. For solutions and candidates with a common regularity bound, secant estimates give linear stability under a relative spectral gap, while covariance interpolation extends residual stability beyond it.