What is wrong with the following simple definition? Let $g_1$ and $g_2$ be two independent standard Gaussian variables. For $t=(a,b)\in S^1$ (so $a^2+b^2=1$), let $B_t=ag_1+bg_2$. You get a Gaussian process whose distribution is invariant under rotation. Each $B_t$ is a standard Gaussian variable and the variance of $B_t-B_s$ is $\|t-s\|^2$.