Stochastic convolution ($Z(t)={\int }_0^{t}S(t-s)Qd{W}_s$)

A stochastic convolution propagates random forcing through a linear evolution. Let $S(t)$ be a strongly continuous semigroup on a Hilbert space $H$, let $W$ be a Cylindrical Wiener process on a Hilbert space $U$, and let $Q:U\to H$ be a bounded linear operator. The expressionis an Itô integral whenever its integrand is square integrable in the Hilbert-Schmidt operator norm. In particular,This Itô isometry tests whether the accumulated forcing is actually $H$-valued. The factor $S(t-s)$ evolves the contribution arriving at time $s$ through the remaining time $t-s$. For $du=Audt+Qd{W}_t$ with deterministic initial value $u_0$, the mild solution is $u(t)=S(t)u_0+Z(t)$.^[ref]

For $S(t)=e^{-at}$, $Q=\sigma$ and real Brownian motion $B$, with $a>0$,This is the zero-initial-value Ornstein-Uhlenbeck process: damping limits the variance despite continued forcing. With $a=0$, the limit is ${\sigma }^{2}t$ instead. These formulas follow by integrating ${\sigma }^2{e}^{-2a(t-s)}$ in the Itô isometry.

The word convolution describes the dependence on the time difference $t-s$. It does not mean an ordinary integral against a differentiable noise function; Brownian motion requires a stochastic integral.