An infinite-dimensional Skorokhod problem with a diagonalization approach
Abstract
We study an infinite-dimensional Skorokhod problem on the positive cone of L1([0,1]), where a regulator process constrains a free càdlàg path to remain nonnegative through an oblique reflection mechanism determined by a positive linear integral operator F on L1([0,1]) with spectral radius less than $1$. This constrained process Z remains in the positive cone, and the regulator Y is nondecreasing with respect to the cone order and satisfies a pointwise complementarity condition, increasing only on those space-time regions where the corresponding component of Z vanishes. This formulation extends the classical multidimensional Skorokhod problem on the nonnegative orthant to an infinite-dimensional setting. Under suitable assumptions on the operator F that guarantee a dominant positive eigenfunction, we proceed through a diagonalization argument to give an alternative representation of the system in which the L1 norm of the linear operator is less than 1. From this characterization, we establish existence and uniqueness of solutions for arbitrary input paths X∈ D([0,T],L1([0,1])). We further show that the associated Skorokhod map is well defined and Lipschitz continuous with respect to both the uniform and Skorokhod J1 topologies. As a consequence, solutions may be obtained as limits of multidimensional Skorokhod problems, providing a constructive characterization of the infinite-dimensional reflection mechanism. These results provide a framework for the study of constrained stochastic systems with infinitely many interacting components and provide a foundation for reflected stochastic processes evolving in function spaces.
Related
- Learning Informative Prior with Infinite-Dimensional Continuous Normalizing Flow for Bayesian Inverse Problem
- Explicit diagonalization of the linearized Calderón problem on the disk: multiplicity normalization and exponential spectral attenuation
- Solving the Tikhonov Regularization of the Total Least Squares Problem by matrix simultaneous diagonalization via congruence
- Diagonalization of nonlinear functions in finite-dimensional spaces and the case of ℝ2
- Formal solution and strong asymptotic expansion of some initial value problem in two complex time variables, infinite order irregular singularities and Mahler transforms
BibTeX
@misc{clarke2026infinitedimensional,
title = {{An infinite-dimensional Skorokhod problem with a diagonalization approach}},
author = {Louis T. Clarke and Guodong Pang and Ruoyu Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03541},
eprint = {2610.03541},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03541},
}
RIS
TY - UNPB
TI - An infinite-dimensional Skorokhod problem with a diagonalization approach
AU - Louis T. Clarke
AU - Guodong Pang
AU - Ruoyu Wu
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03541
UR - https://arxiv.org/abs/2610.03541
AB - We study an infinite-dimensional Skorokhod problem on the positive cone of L_1([0,1]), where a regulator process constrains a free càdlàg path to remain nonnegative through an oblique reflection mechanism determined by a positive linear integral operator F on L_1([0,1]) with spectral radius less than $1$. This constrained process Z remains in the positive cone, and the regulator Y is nondecreasing with respect to the cone order and satisfies a pointwise complementarity condition, increasing only on those space-time regions where the corresponding component of Z vanishes. This formulation extends the classical multidimensional Skorokhod problem on the nonnegative orthant to an infinite-dimensional setting. Under suitable assumptions on the operator F that guarantee a dominant positive eigenfunction, we proceed through a diagonalization argument to give an alternative representation of the system in which the L_1 norm of the linear operator is less than 1. From this characterization, we establish existence and uniqueness of solutions for arbitrary input paths X∈ D([0,T],L_1([0,1])). We further show that the associated Skorokhod map is well defined and Lipschitz continuous with respect to both the uniform and Skorokhod J_1 topologies. As a consequence, solutions may be obtained as limits of multidimensional Skorokhod problems, providing a constructive characterization of the infinite-dimensional reflection mechanism. These results provide a framework for the study of constrained stochastic systems with infinitely many interacting components and provide a foundation for reflected stochastic processes evolving in function spaces.
ER -