Reflected brownian motion: selection, approximation and linearization

TL;DR

This paper constructs a family of stochastic differential equations that select reflected Brownian flows and provides a representation for solutions to the heat equation for differential 1-forms with boundary conditions, including approximation results for boundary local time.

Contribution

It introduces a novel family of SDEs that select reflected Brownian flows and offers a new representation for heat equation solutions with boundary conditions, including approximation methods.

Findings

Constructs SDEs that select reflected Brownian flows.

Provides a representation for heat equation solutions for differential 1-forms.

Develops an approximation for boundary local time in uniform topology.

Abstract

We construct a family of SDEs whose solutions select a reflected Brownian flow as well as a stochastic damped transport process (W\_t). The latter gives a representation for the solutions to the heat equation for differential 1-forms with the absolute boundary conditions; it evolves pathwise by the Ricci curvature in the interior, by the shape operator on the boundary and driven by the boundary local time, and has its normal part erased on the boundary. On the half line this construction selects the Skorohod solution (and its derivative with respect to initial points), not the Tanaka solution. On the half space this agrees with the construction of N. Ikeda and S. Watanabe \cite{Ikeda-Watanabe} by Poisson point processes. This leads also to an approximation for the boundary local time in the topology of uniform convergence; not in the semi-martingale topology, indicating the difficulty for the convergence of solutions of a family of random ODE's, with nice coefficients, to the solution of an equation with jumps and driven by the local time. In addition, We note that (W\_t) is the weak derivative of a family of reflected Brownian motions with respect to the starting point.