Cylindrical Brownian motion ($\mathbb{E}[W_t(h)W_s(k)]=(t\wedge s)⟨h,k⟩$)

On a real Hilbert space $H$, this centered jointly Gaussian process is linear in its test vector $h$ and has the displayed covariance, with independent increments in time. Equivalently choose an orthonormal basis and independent real Brownian motions $W^j$, and define stochastic integration by $\int ⟨v_s,d{W}_s⟩={\sum }_j\int ⟨v_s,e_j⟩d{W}_s^j$ whenever $\int \Vert v_s{\Vert }^{2}ds<\infty$. In infinite dimension it need not be an $H$-valued random variable; the integral is the meaningful object.