  We generalize the classical Bochner formula for the heat flow on evolving
manifolds $(M,g_{t})_{t \in [0,T]}$ to an infinite-dimensional Bochner formula
for martingales on parabolic path space $P\mathcal{M}$ of space-time
$\mathcal{M} = M \times [0,T]$. Our new Bochner formula and the inequalities
that follow from it are strong enough to characterize solutions of the Ricci
flow. Specifically, we obtain characterizations of the Ricci flow in terms of
Bochner inequalities on parabolic path space. We also obtain gradient and
Hessian estimates for martingales on parabolic path space, as well as condensed
proofs of the prior characterizations of the Ricci flow from Haslhofer-Naber
\cite{HN18a}. Our results are parabolic counterparts of the recent results in
the elliptic setting from \cite{HN18b}.
