On a multidimentional Brownian motion with partly reflecting membrane on a hyperplane

TL;DR

This paper constructs a multidimensional Brownian motion with partial reflection on a hyperplane, analyzing its distribution and local time using both analytic and probabilistic methods.

Contribution

It introduces a new construction of multidimensional Brownian motion with partial reflection on a hyperplane, combining analytic and probabilistic approaches.

Findings

Derived the joint distribution of the process and its local time.

Established a construction method for the reflected Brownian motion.

Analyzed the process's behavior with respect to the hyperplane.

Abstract

A multidimensional Brownian motion with partial reflection on a hyperplane $S$ in the direction $qN+\alpha $, where $N$ is the conormal vector to the hyperplane and $q\in [-1,1], \alpha \in S$ are given parametres, is constructed and this construction is based on both analytic and probabilistic approaches. The joint distribution of $d-$dimensional analogy to skew Brownian motion and its local time on the hyperplane is obtained.