Gaussian covariance differentiation identity (${\partial }_t{\mathbb{E}}_{C_t}W={\mathbb{E}}_{C_t}[{\partial }_{t}W+\frac{1}{2}{\dot{C}}_{ab}{\partial }_a{\partial }_{b}W]$)

For a positive differentiable covariance $C_t$ and a sufficiently regular, integrable $W$, differentiating the normalized Gaussian integral and integrating by parts twice gives the displayed identity. An unnormalized integral has the additional field-independent term $\mathrm{Tr}(C_t^{-1}{\dot{C}}_t)/2$ times the integral. A finite-mode regulator makes the analogous Gaussian functional integral identity precise before taking a continuum limit.