Heat semi-group and generalized flows on complete Riemannian manifolds (Q645938)

The aim of this paper is to prove the existence and uniqueness on a Riemannian manifold \(M\) of a flow solution of \(dX_t(x)/dt=V_t(X_t(x))\) when the vector field \(V\) has only Sobolev regularity. This problem has been studied by R. J.~Di Perna and P. L.~Lions when \(M\) is an Euclidean space. One has to regularize \(V\), and here, this is done by means of the heat semiflow \(T_t^M\). By means of stochastic methods (and therefore the Brownian motion on \(M\)), the authors prove some estimates for the commutator between \(T_t^M\) and the Lie derivative \(L_V\) along \(V\), and apply these estimates to conclude the results. In the last section, they also construct a diffusion with drift \(V\).