Brownian motion on the circle and applications (Q2866254)

The authors give an explicit representation of the \(1\)-dimensional reflecting Brownian motion on an interval in terms of a free \(1\)-dimensional Brownian motion on the circle, similar to the Tanaka formula.NEWLINENEWLINE As an application, the authors derive a probabilistic proof of the Laugesen-Morpurgo conjecture regarding the monotonicity of the trace of the Neumann heat kernel.NEWLINENEWLINE The interest for explicit representations of reflecting Brownian motion comes from various problems concerning reflecting Brownian motion: bounds on the transition density of the reflecting process, open problems concerning reflecting Brownian motion, e.g., the hot spots conjecture by J. Rauch, Isaac Chavel's conjecture on the domain monotonicity of the transition density of the reflecting Brownian motions, the Laugesen-Morpurgo conjecture on the monotonicity of the trace of the Neumann heat kernel for the unit ball in \(\mathbb R^n\), etc.NEWLINENEWLINE The representation obtained in this paper solves the problem in the \(1\)-dimensional case, by extending Tanaka's formula to an arbitrary interval in \(\mathbb R\), and it gives insight for a possible approach of the general case of smooth domains \(D\) in \(\mathbb R^n\).