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 diffusion marginals. The reference diffusion determines the norm, and its generator may have different principal coefficients from the PDE. We characterize the limiting optimal Hessian constant uniformly over shrinking subintervals of a fixed time horizon, with zero terminal data. Under coefficient regularity, uniform ellipticity and bounded reference drift, finiteness characterizes linear well-posedness in the weighted parabolic Sobolev space on the full interval. For initial laws satisfying weighted doubling and coercivity conditions, the constant is the maximum of initial, flat and exponential-mixture contributions. Additional geometric regularity and nonnegative Ricci curvature identify the full mixture contribution, including spatial infinity, through terminal momenta of paths minimizing action plus the initial potential. In the matching case, with reference covariance $a$, PDE matrix $a/2$ and normalized Hessian $a^{1/2}D^2u\,a^{1/2}$, the constant is $2$ with time weight $t^\alpha$, $\alpha\ge1/2$, for every initial law; point starts attain $2\sqrt2$ at $\alpha=0$.