REFLECTED BROWNIAN MOTION: SELECTION, APPROXIMATION AND LINEARIZATION
Résumé
We construct a family of SDEs whose solutions select a reflected Brownian flow as well as a stochastic damped transports process, the latter solves the heat equation for differential 1-forms with absolute boundary conditions. The damped transport process evolves pathwise by the Ricci curvature in the interior, by the shape operator on the boundary driven by the boundary local time, and has its normal part erased on the boundary. On the half line this selects the Skorohod solution, not the Tanaka solution, and its derivative with respect to initial points.
On the half space this agrees with the construction in N. Ikeda and S. Watanabe \cite{Ikeda-Watanabe} by Poisson point processes. This gives also an approximation for the boundary local time in the topology of uniform convergence, but not in the semi-martingale topology, indicate the difficulty for for the convergence of solutions to random ODE with nice coefficients to the solution of an equation with jumps and driven by the local time. We also conclude $W_t$ is the weak derivative of a family od reflected Brownian motion with respect to the starting point.
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