Complete lifts of connections and stochastic Jacobi fields

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Abstract

Differentiable 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.

MSC

58G32
60H30

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