The purpose of the author is the study of Brownian motion \((X_t)\) on a Riemannian manifold \(M\) whose metric \(g(t)\) is also evolving in time (deterministically). Precisely, \((X_t)\) is the unique solution to the martingale problem associated with the Laplace-Beltrami operator \(\Delta(t)\) arising from \(g(t)\). Its density satisfies a perturbed heat equation, in which \(g'(t)\) appears.
Then the author introduces, for fixed \(T> 0\), the reversed Brownian motion \((X^T_t)\) associated to \(g(T- t)\), and an associated damped parallel transport \((W^T_t)\), which is an isometry enjoying the following property: if \(g(t)\) is a Ricci flow, then \((W^T_t)\) coincides with the usual parallel transport.
If \(f\) satisfies the perturbed heat equation, then \(df(T- t_j)_{X^T}(W^T_t)\) is a local martingale. A Bismut formula is deduced from that fact, and then gradient estimates follows.
Maybe the main result is the following stochastic characterization: \(g(t)\) is a Ricci flow if and only if the parallel transport is given by the differential \(T_x X^T_t(x)\) of the Brownian flow \(X^T_t(x)\).
Finally some intrinsic \(TM\)-valued martingale is brought out, namely \((//^T_t)^{-1}\text{Tr\,}\nabla T\,X^T_t(x)\), arising from the Hessian of the Brownian flow and having quadratic variation \[ \|\text{Ricci}_{T-t}(X^T_t(x))\|^2_{g(T- t)}. \]