= Stochastic convolution
{title2=$Z(t)=\int_0^tS(t-s)Q\,dW_s$}
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 expression
$$
Z(t)=\int_0^tS(t-s)Q\,dW_s
$$
is an <Itô integral> whenever its integrand is square integrable in the <Hilbert-Schmidt operator> norm. In particular,
$$
\mathbb E\|Z(t)\|_H^2=\int_0^t\|S(t-s)Q\|_{\mathrm{HS}}^2\,ds<\infty.
$$
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 $du=Au\,dt+Q\,dW_t$ with deterministic initial value $u_0$, the <mild solution> is $u(t)=S(t)u_0+Z(t)$.<An Introduction to Stochastic PDEs by Martin Hairer>[§6, Definition 6.5, equation 6.3 and Remark 6.6; §4.4, Hilbert-space stochastic integration]{ref}
For $S(t)=e^{-at}$, $Q=\sigma$ and real <Brownian motion> $B$, with $a>0$,
$$
Z(t)=\sigma\int_0^te^{-a(t-s)}\,dB_s,\qquad\operatorname{Var}Z(t)=\frac{\sigma^2}{2a}(1-e^{-2at}).
$$
This is the zero-initial-value <Ornstein-Uhlenbeck process>: damping limits the <variance> despite continued forcing. With $a=0$, the limit is $\sigma^2t$ instead. These formulas follow by integrating $\sigma^2e^{-2a(t-s)}$ 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>.
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