= Solution
The integrated <stochastic differential equation> reads
$$
Y(t):=y(t)-y(0)=\int_0^t\eta(s)\,ds.
$$
The <Gaussian white noise> has zero <expected value>, so
$$
\boxed{\langle y(t)-y(0)\rangle=0.}
$$
Its <Dirac delta> <covariance> gives
$$
\begin{aligned}
\langle Y(t)^2\rangle
&=\int_0^t\!ds\int_0^t\!ds'\,\langle\eta(s)\eta(s')\rangle\\
&=\Gamma\int_0^t\!ds\int_0^t\!ds'\,\delta(s-s')
=\Gamma t.
\end{aligned}
$$
Thus $\boxed{\langle[y(t)-y(0)]^2\rangle=\Gamma t}$, which is also the <variance> because the displacement has zero <expected value>. More generally its <covariance> is $\langle Y(s)Y(t)\rangle=\Gamma\min(s,t)$. Equivalently $Y(t)=\sqrt\Gamma\,W_t$ for standard <Brownian motion>, and the <diffusion coefficient> is $D=\Gamma/2$.
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