Brownian sheet ($\mathrm{Cov}(W(t,T),W(s,U))=\min (t,s)\min (T,U)$)

A centered Gaussian random field with the displayed covariance. It has independent increments over disjoint rectangles. At fixed positive $T$, $W(t,T)/\sqrt{T}$ is standard Brownian motion in $t$. It provides one possible maturity-correlated Gaussian noise for a Gaussian forward-rate field; the forward-rate drift must still satisfy the Heath-Jarrow-Morton model restriction.