An infinite-dimensional Skorokhod problem with a diagonalization approach

Abstract

We study an infinite-dimensional Skorokhod problem on the positive cone of $L_1([0,1])$, where a regulator process constrains a free c\`adl\`ag path to remain nonnegative through an oblique reflection mechanism determined by a positive linear integral operator $\boldsymbol{F}$ 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 $\boldsymbol{F}$ 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 $X\in D([0,T],L_1([0,1]))$. 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.