Complete lifts of connections and stochastic Jacobi fields

Abstract

Differentiable families of V-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 V. We discuss various characterizations of TM-valued martingales. When applied to specific families of V-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. 0 Elsevier, Paris

Horizontal martingales in vector bundles

“…Extending our work in the case Ì Å [2], we focus here on various properties of horizontal and complete lifts, in particular in relation to the variation of families of semimartingales. We differentiate martingales with respect to a parameter, take exterior products, to obtain martingales for complete lifts of connections.…”

“…Theorem 2.2 ( [3] Corollary 3.14) guarantees that Itô stochastic differential equations between manifolds differentiate like ordinary equations, if the tangent bundles are equipped with the complete lifts of the connections in the manifolds. Corollary 2.3 ([3] Theorem 4.1 and [2] Theorem 3.1) gives a commutation formula between antidevelopment and differentiation with respect to a parameter, and an interpretation of martingales in the tangent bundle as stochastic Jacobi fields, i.e. as derivatives of families of martingales living in the manifold.…”

“…For (i) and (ii), we may follow the proof of Theorem 3.1 in [2]; the difference is that Stratonovich differentials are replaced by Itô differentials, and the crucial result for differentiation with respect to a parameter is now Theorem 2.2.…”

Yang--Mills fields and random holonomy along Brownian bridges

“…the connection ∇ in E. Let J be a continuous E-valued semimartingale and X = π •J . As shown in [2], the antidevelopment of J with respect to ∇ h is given by the formula…”

“…In fact, the characterizations of Theorem 2.1 and Corollary 2.2 equally hold for geodesically complete noncompact manifolds, if stated with local martingales. 2 and X(a) = X(a, u) be a rescaled Brownian bridge from exp x 0 (au 1 ) to exp x 0 (au 2 ) with lifetime 1 defined as follows:…”