PDE Generators from Energy Space Structure in Feynman-Kac Theory

The operator factorization framework for stochastic calculus defines an operator derivative DX via adjointness to the stochastic integral, with energy function GammaX (t) = ||Pi DX Xₜ||² replacing classical quadratic variation. This paper asks: what does the energy space H determine about the PDE structure of conditional expectations? The answer is a structural principle, not a proof technique. For Gaussian processes, GammaX (t) = R (t, t) is deterministic, and the marginal density PDE has coefficient (1/2) d/dt GammaX (t). This recovers classical diffusion operators for Brownian motion and time-dependent diffusivities for fractional Brownian motion, and proves that continuous Gaussian memory can only scale the time derivative — never producing a spatial fractional operator. For iterated Brownian motion, the tensor-product energy space produces a biharmonic generator; we derive the complete distributional PDE: the time derivative of p equals (1/8) times the fourth spatial derivative of p, plus a local-time source term concentrated at the origin. For Levy processes, the jump energy space L² (dt x nu) produces nonlocal operators including the fractional Laplacian. Every PDE derived in this paper can also be obtained by direct methods — density calculations for Gaussian processes, conditional analysis and Fourier methods for iterated Brownian motion, compensator calculations for Levy processes. The operator framework does not replace these proofs; it explains, after the fact, why each energy space produces its particular PDE structure. The genuine proof-mechanical contributions of the operator factorization program — including the cylindrical reduction principle, the Leibniz obstruction for stochastic volatility, and the representability obstruction for jump processes — are developed in the companion papers. The present paper serves as the PDE-side application: it shows that the energy space classification has concrete analytical consequences.