A Bochner formula on path space for the Ricci flow

We generalize the classical Bochner formula for heat flow on evolving manifolds $$(M,g_{t})_{t \in [0,T]}$$ to an infinite-dimensional martingales parabolic path space $$P{\mathcal {M}}$$ of space-time $${\mathcal {M}} = M \times [0,T]$$ . Our new and inequalities that follow from it are strong enough characterize solutions Ricci flow. Specifically, we obtain characterizations in terms space. also gradient Hessian estimates space, as well condensed proofs prior Haslhofer–Naber (J Eur Math Soc 20(5):1269–1302, 2018a). results counterparts recent elliptic setting (Commun Pure Appl 71(6):1074–1108, 2018b).