= Solution
The density $Z=\exp(X(k)-\|k\|^2/2)$ is strictly positive. The <Gaussian moment-generating function> gives $\mathbb E Z=1$, so it defines an <equivalent probability measure>.
The joint <normal distribution> of $(X(h),X(k))$, with <covariance> $\langle h,k\rangle$, gives the mixed exponential formula
$$
\mathbb E_{\mathbb P}e^{X(k)+i\theta X(h)}=\exp\left(\frac12\|k\|^2+i\theta\langle h,k\rangle-\frac12\theta^2\|h\|^2\right).
$$
Multiplying by the normalizing and centring factors therefore yields
$$
\mathbb E_{\mathbb Q}e^{i\theta(X(h)-\langle h,k\rangle)}=\exp\left(-\frac12\theta^2\|h\|^2\right).
$$
The <characteristic function> identifies the answer:
$$
\boxed{X(h)-\langle h,k\rangle\sim N(0,\|h\|^2)\quad\text{under }\mathbb Q.}
$$
This is <exponential tilting of an isonormal Gaussian process>: the mean shifts by the inner product while its <covariance> remains unchanged.
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