Let $\mathsf{BPTIME}(f(n))$ be the class of the decision problems having a bounded two-sided error randomized algorithm running in time $O(f(n))$.
Do we know of any problem $Q \in \mathsf{P}$ such that $Q \in \mathsf{BPTIME}(n^k)$ but $Q \not \in \mathsf{DTIME}(n^k)$? Is its non-existence proven?
This question was asked on cs.SE here, but did not get a satisfactory answer.