= Solution
Fix $t<\infty$. Part (b)(i) makes $d\mathbb Q_t=Z_t\,d\mathbb P$ a <probability measure> on $\mathcal F_t$, equivalent to $\mathbb P$ since $Z_t>0$. Its density process on $[0,t]$ is $Z_s=\mathbb E[Z_t\mid\mathcal F_s]$. The <Girsanov theorem>, applied to the <stochastic exponential> with integrand $-\beta B_s$, says that
$$
W_s=B_s+\beta\int_0^sB_u\,du,\qquad 0\leq s\leq t,
$$
is standard <Brownian motion> under $\mathbb Q_t$. The required local square-integrability follows from path continuity, and the true <martingale> property was proved in part (b)(i).
By part (b)(ii), under $\mathbb Q_t$ the process $B$ is therefore the zero-started <Ornstein-Uhlenbeck process>:
$$
\boxed{dB_s=dW_s-\beta B_sds,\qquad
B_s=\int_0^se^{-\beta(s-u)}\,dW_u,\quad 0\leq s\leq t.}
$$
This identifies its full law on continuous path space, rather than just its one-time marginal. It is a centered <Gaussian process>, since it consists of deterministic <Itô integrals> against <Brownian motion>. For $r,s\in[0,t]$, the <Itô isometry> gives its <covariance function>:
$$
\boxed{\operatorname{Cov}_{\mathbb Q_t}(B_r,B_s)
=\int_0^{r\wedge s}e^{-\beta(r-u)}e^{-\beta(s-u)}du
=\frac{e^{-\beta|s-r|}-e^{-\beta(s+r)}}{2\beta}.}
$$
In particular, its <Ornstein-Uhlenbeck transition distribution> at the terminal time is
$$
B_t\sim N(0,v_t),\qquad v_t=\frac{1-e^{-2\beta t}}{2\beta}.
$$
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