Understanding the complexity class $P^O$ for randomized oracles

Question

We know from Toda's theorem that $PH \subseteq P^{PP}$. What do we know about the following classes?

$$ P^{ZPP}, P^{RP}, \text{ and } P^{BPP} $$

Answer 1

$\let\mr\mathrm \mr{BPP}$ is in the second level of polynomial hierarchy by the Gács–Sipser–Lautemann theorem, and it is an easy exercise to show that $$\mr{P^{ZPP}}=\mr{ZPP}\subseteq\mr{P^{RP}}\subseteq\mr{P^{BPP}}=\mr{BPP}.$$ The least trivial of these is the inclusion $\mr{P^{BPP}}\subseteq\mr{BPP}$: let $L_0$ be a language decidable by a machine $M_0$ running in time $n^c$ with an oracle for a language $L_1\in\mr{BPP}$. By standard amplification, $L_1$ is computable by a randomized poly-time machine $M_1$ with error bounded by (say) $2^{-n}$. So, if we take $M_0$, and replace the oracle with $M_1$ as a subroutine, we obtain a randomized poly-time machine for $L_0$ with error bounded by $n^c2^{-n}<1/3$.

$\mr{P^{RP}}$ contains $\mr{RP}\cup\mr{coRP}$, hence it equals $\mr{RP}$ only if $\mr{RP}=\mr{coRP}=\mr{ZPP}$. This is not known to be true, nevertheless all the classes mentioned above are conjectured to coincide with $\mr P$, and there are conditional results to that effect.