We constructed a family of SDEs with parameter a whose solutions converge to a reflected Brownian flow as a approaches 0 in UCP. This selects a reflected Brown-ian 'flow'. We then prove the stochastic damped transports W a t along (Y a t) converge in a suitable sense. This selects a damped stochastic parallel transport and a probabilistic representation for the solution to the heat equation on differential 1-forms with absolute boundary conditions. The limiting process can be described by its jumps and the standard stochastic damped parallel transport equation with an additional shape operator driven by the local time. It can be constructed by a family of tangent space valued stochastic processes (W ε t), who evolves with the stochastic damped parallel transport equation in the interior, and whose normal part is erased at the end of an excursion of size greater or equal to ǫ. As ǫ → 0, this family of stochastic processes converge to a tangent space valued process (Wt) with jumps, in the topology of uniform convergence. For the half space, this is exactly the process constructed in N. Ikeda and S. Watanabe [21] by Poisson point processes. This allow us to approximate the local time on the boundary in UCP, but not in the semi-martingale topology. We also conclude Wt being the weak derivative of Yt with respect to the starting point.