in DaPrato/Zabczyk's book "Second Order Partial Differential Equations in Hilbert Spaces", there is a useful proposition (Prop. 1.2.8) about a particular calculation of a Gaussian integral in Hilbert spaces: Take a symmetric operator $M$ in a Hilbert space, a centred Gaussian measure $N\_Q$ on $H$ with covariance operator $Q$ and assume $ \langle Q^\frac{1}{2} MQ^\frac{1}{2}u,u\rangle < \langle u , u\rangle$ for all $u\neq 0$. Then for $b\in H$,


$$\int\_H\exp\left\{\frac{1}{2}\langle M y,y\rangle + \langle b,y\rangle\right \}N\_Q(d y) = \frac{\exp\left\{\frac{1}{2}|(1-Q^\frac{1}{2}MQ^\frac{1}{2})^{-\frac{1}{2}}Q^\frac{1}{2}b|^2\right\}}{\sqrt{\det (1-Q^\frac{1}{2}MQ^\frac{1}{2})}}.$$


My problem is that I need to do a calculation of the form


$$\int\_H \exp\left\{\frac{1}{2}\langle My,y\rangle + \langle b\_1 + ib\_2, y\rangle\right\}N\_Q(dy).$$


This is almost the setting where I could use the formula above, but it cannot hold in the literal sense, with replacing $b = b\_1 + b\_2 i$, as the left hand side of the formula can (and will be) properly complex, whereas the right hand side is always real.


Does anyone know of a generalization of this formula? I did the calculation in 1d and there I get a *square* on the right hand side (which can be complex) instead of the *squared absolute value* (which will be real).



complex version of Gaussian integral
$$\int\_H \exp\left\{\frac{1}{2}\langle My,y\rangle + \langle b\_1 + ib\_2, y\rangle\right\}N\_Q(dy)=$$
$$= \frac{\exp\left\{\frac{1}{2}\langle b\_1+ib\_2,Q^{1/2}(1-Q^{1/2}MQ^{1/2})^{-1}Q^{1/2}(b\_1+ib\_2)\rangle\right\}}{\sqrt{\det (1-Q^\frac{1}{2}MQ^\frac{1}{2})}}$$


the inner product is defined here without complex conjugation:


$$\langle b\_1+ib\_2,c\rangle=\langle b\_1,c\rangle+i\langle b\_2,c\rangle$$


so the right-hand-side will be complex if $b\_2\neq 0$.


