2024/07/06 by Richard F. Bass, Krzysztof Burdzy, Bass, Richard F. +1
Engineering · Mathematics · #60J65 #Differential Equations and Boundary Problems #Elasticity and Wave Propagation #FOS: Mathematics #Geotechnical and Geomechanical Engineering #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2407.05140
openalex publication_date 2024/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the Skorokhod problem in the closed non-negative orthant: find a solution (g(t),m(t)) to g(t)= f(t)+ Rm(t), where f is a given continuous vector-valued function with f(0) in the orthant, R is a given d× d matrix with 1's along the diagonal, g takes values in the orthant, and m is a vector-valued function that starts at 0, each component of m is non-decreasing and continuous, and for each i the ith coordinate of m increases only when the ith coordinate of g is 0. The stochastic version of the Skorokhod problem replaces f by the paths of Brownian motion. It is known that there exists a unique solution to the Skorokhod problem if the spectral radius of |Q| is less than 1, where Q=I-R and |Q| is the matrix whose entries are the absolute values of the corresponding entries of Q. The first result of this paper shows pathwise uniqueness for the stochastic version of the Skorokhod problem holds if the spectral radius of |Q| is equal to 1. The second result of this paper settles the remaining open cases for uniqueness for the deterministic version when the dimension d is two.