Approximating damped parallel translations along a Reflected Brownian Motion
by Marc Arnaudon (Univ. Bordeaux)
Following N. Ikeda and S. Watanabe and motivated by the problem of solving the heat equation for differential 1-forms, we study damped parallel translation $(W_t)$ along a reflected Brownian motion $(Y_t)$ in a Riemannian manifold with boundary. Before the first hitting time of the boundary, $(W_t)$ is standard damped parallel translation. Upon arriving on the boundary its normal part is removed after each excursion.
It solves an equation with jumps, involving the Ricci curvature, the shape operator and the local time $L_t$ at~$0$ of the distance function to the boundary.
We construct a family of approximating stochastic processes $(Y_t^a, L_t^a, W_t^a)$ where, for each $a$, $(Y_t^a)$ is a smooth stochastic flow stayingin the interior, solving an equation driven by sample continuous stochastic processesand with large drift $A^a$ pushing away from the boundary. The tangent bundle valued processes $(W_t^a)$ are the standard stochastic parallel transports damped by the Ricci curvature and covariant derivative of $A^a$, equivalently they are the conditioned derivativeflows obtained from differentiating the stochastic flows with respect to the initial conditions. The approximation procedure allows to see $(W_t)$ as weak derivative of $(Y_t)$ with respect to starting point.