Complete lifts of connections and stochastic Jacobi fields

Differentiable families of del-martingales on manifolds are investigated: their infinitesimal variation provides a notion of stochastic Jacobi fields. Such objects are known [2] to be martingales taking values in the tangent bundle when the latter is equipped with the complete lift of the connection del. We discuss various characterizations of TM-valued martingales. When applied to specific families of del-martingales which appear in connection with the heat how for maps between Riemannian manifolds, our results allow to establish formulas giving a stochastic representation for the differential of solutions to the nonlinear heat equation. As an application, we prove local and global gradient estimates for harmonic maps of bounded dilatation. (C) Elsevier, Paris.