发表机构
Rice University; Iowa State University（莱斯大学; 爱荷华州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究$L_1([0,1])$正锥上的无穷维Skorokhod问题，通过对角化方法将系统转化为$L_1$范数小于1的等价形式，证明了解的存在唯一性及Skorokhod映射的Lipschitz连续性，为无穷维反射随机过程奠定基础。
AI 中文摘要
我们研究了$L_1([0,1])$正锥上的一个无穷维Skorokhod问题，其中，一个调节过程通过由$L_1([0,1])$上谱半径小于1的正线性积分算子$\boldsymbol{F}$确定的斜反射机制，约束一个自由的càdlàg路径保持非负。该受约束过程$Z$保持在正锥内，且调节器$Y$关于锥序是非减的，并满足逐点互补条件，仅在$Z$相应分量消失的时空区域上增加。该表述将经典的非负象限上的多维Skorokhod问题推广到无穷维情形。在对算子$\boldsymbol{F}$保证存在主导正特征函数的适当假设下，我们通过对角化论证给出系统的一个替代表示，其中线性算子的$L_1$范数小于1。由此刻画，我们建立了对任意输入路径$X\in D([0,T],L_1([0,1]))$解的存在唯一性。我们进一步证明了相应的Skorokhod映射是良定义的，并且在一致拓扑和Skorokhod $J_1$拓扑下都是Lipschitz连续的。因此，解可以作为多维Skorokhod问题的极限获得，从而提供了无穷维反射机制的构造性刻画。这些结果为研究具有无穷多个相互作用分量的受约束随机系统提供了框架，并为函数空间中反射随机过程的研究奠定了基础。
英文摘要
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 $\boldsymbol{F}$ 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.
