= Gaussian covariance differentiation identity
{c}
{title2=$\partial_t\mathbb E_{C_t}W=\mathbb E_{C_t}[\partial_tW+\tfrac12\dot C_{ab}\partial_a\partial_bW]$}
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 $\operatorname{Tr}(C_t^{-1}\dot C_t)/2$ times the integral. A finite-mode regulator makes the analogous <Gaussian functional integral> identity precise before taking a continuum limit.
Back to article page
