Reflected Brownian motion in a cone with radially homogeneous reflection field

Abstract:

This work is concerned with the existence and uniqueness of a strong Markov process that has continuous sample paths and the following additional properties. (i) The state space is a cone in $d$-dimensions $(d \geq 3)$, and the process behaves in the interior of the cone like ordinary Brownian motion. (ii) The process reflects instantaneously at the boundary of the cone, the direction of reflection being fixed on each radial line emanating from the vertex of the cone. (iii) The amount of time that the process spends at the vertex of the cone is zero (i.e., the set of times for which the process is at the vertex has zero Lebesgue measure). The question of existence and uniqueness is cast in precise mathematical terms as a submartingale problem in the style used by Stroock and Varadhan for diffusions on smooth domains with smooth boundary conditions. The question is resolved in terms of a real parameter $\alpha$ which in general depends in a rather complicated way on the geometric data of the problem, i.e., on the cone and the directions of reflection. However, a criterion is given for determining whether $\alpha > 0$. It is shown that there is a unique continuous strong Markov process satisfying (i)-(iii) above if and only if $\alpha < 2$, and that starting away from the vertex, this process does not reach the vertex if $\alpha \leq 0$ and does reach the vertex almost surely if $0 < \alpha < 2$. If $\alpha \geq 2$, there is a unique continuous strong Markov process satisfying (i) and (ii) above; it reaches the vertex of the cone almost surely and remains there. These results are illustrated in concrete terms for some special cases. The process considered here serves as a model for comparison with a reflected Brownian motion in a cone having a nonradially homogeneous reflection field. This is discussed in a subsequent work by Kwon.