Complete lifts of connections and stochastic Jacobi fields

AbstractDifferentiable families of ∇-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 ∇. We discuss various characterizations of TM-valued martingales. When applied to specific families of ∇-martingales which appear in connection with the heat flow 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