Local Fokker--Planck Geometry for Score Estimation: Heat-Ball Mean-Value Representations and Exact High-Dimensional Sampling

Score-based generative models and Langevin samplers rely on estimating the score function $\nabla_x\log p_t(x)$ of a forward diffusion.

Classically this is tractable when the drift is linear: the marginal density is Gaussian and the score is a global conditional expectation.

For a general nonlinear, state-dependent drift the marginal density has no closed form, and existing methods--denoising score matching and global Fokker--Planck residual penalties--resort to global averaging that inflates estimation error in low-density regions precisely where accuracy is most critical.

We address this by developing a local Fokker--Planck geometric framework that replaces global conditioning with local parabolic averaging.

Our approach rests on three contributions.

First, a time change to the cumulative-variance coordinate reduces the variable-coefficient Fokker--Planck equation to a standard inhomogeneous heat equation, on which we extend Evans' classical heat-ball monotonicity method to derive exact local mean-value representations for the score $\nabla_x\log p$ together with the density, log-density, and entropy density; local well-posedness is established under an explicit dimension-dependent drift budget.

Second, for high-dimensional Monte Carlo evaluation of the resulting heat-ball integrals, we introduce the $\kappa$-measure and derive its exact factorized sampler with unit per-sample weight, $\chi^2_2$ radial concentration.

Third, the $r\to0$ limit of the heat-ball residual recovers the pointwise Fokker--Planck residual, showing that the local framework is a one-parameter generalization of global FP-residual methods, and that the DSM population minimizer is feasible for the heat-ball constraint at every scale.

We validate the framework on 2D structured data on 256-dimensional MNIST, and on a dedicated sampler study confirming the concentration laws.