Suppose $B_1(t)$ and $B_2(t)$ are two independent, standard Brownian motions. What is the distribution of \begin{align*} G(t) = \int_0^t B_1(\tau)B_2(t-\tau)d\tau \qquad \end{align*} Or, at least an approximation to $\text{Var}(G(t))$?

**Generalizations**

(1) How about $B_1, B_2$ being any Gaussian process with covariance functions $c_1(t, s)$ and $c_2(t, s)$?

(2) How about \begin{align*} G(t) = \int_0^t B_2(t-\tau)dB_1(\tau) \qquad \end{align*} For either $B_1, B_2$ being Brownian motions or, more generally, Gaussian processes?