Let $\mathcal{C}$ be a complexity class and $\textrm{BP-}\mathcal{C}$ be the randomized counterpart of $\mathcal{C}$ defined as $\textrm{BPP}$ with respect to $\textrm{P}$. More formally we provide polynomially many random bit and we accept an input iff the probability to accept is over $\frac{2}{3}$.
It is known that for non-uniform circuits class we have $\textrm{BPAC}^0=\textrm{AC}^0$:
Miklós Ajtai, Michael Ben-Or: A Theorem on Probabilistic Constant Depth Computations STOC 1984: 471-474
Are generalizations of this theorem known ? For instance, do we know if $\mathrm{BPNC}^1=\mathrm{NC}^1$ (still in the non-uniform setting)? This last question seems somehow non trivial to me since it seems plausible that for instance $s,t\textrm{-Connectivity}$ is in $\textrm{BPNC}^1$.
A relevant post on the subject: https://mathoverflow.net/questions/35184/use-of-randomness-in-constant-parallel-time