发表机构
University of Ottawa; University of California San Diego; University of Michigan（渥太华大学; 加州大学圣迭戈分校; 密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
我们提出非交换$L^p$随机积分新框架，定义可预测线性过程，建立积分算子判据，应用于$L^p$中SDE，证明极大解存在唯一性及爆破性，推广自由和$q$-布朗运动SDE结果。
AI 中文摘要
我们提出了一个受 Bichteler 的 $L^p$-积分算子概念启发的非交换随机积分新框架，该框架极大地推广了我们先前关于针对 $L^2$-可分解过程的 $L^2$-值随机积分的理论。我们框架的核心是一种可预测线性过程，它是可预测被积函数的非交换类比。我们发展了针对“非交换 $L^p$-积分算子”的随机积分的基本性质，并建立了一个有用的判据，用以判定非交换随机过程是否为非交换 $L^p$-积分算子。该判据尤其适用于一类我们称之为可测分解过程的丰富过程类，其中包括自由布朗运动，更一般地，还包括 $q$-布朗运动。我们还证明，在适当特化时，我们的框架恢复了经典 $L^p$-积分算子的概念，但非交换理论仅关注经典随机过程的修正类。作为应用，我们利用我们的理论研究 $L^p$ 中的非交换随机微分方程（SDE）。在自然的局部 Lipschitz 和有界性假设下，我们建立了极大解的存在唯一性，并证明了具有有限寿命的极大解必须在 $L^p$ 范数下爆破。作为推论，我们获得了自由 SDE 和由 $q$-布朗运动驱动的 SDE 的新的存在唯一性结果。值得注意的是，在自由情形下，我们的结果适用于通过连续泛函演算从仅局部 Lipschitz 标量函数（而非局部算子-Lipschitz 函数）导出的系数。
英文摘要
We propose a new framework for noncommutative stochastic integration in $L^p$, inspired by Bichteler's notion of $L^p$-integrators, that massively generalizes our previous theory of $L^2$-valued stochastic integration against $L^2$-decomposable processes. Central to our framework is a noncommutative analog of a predictable integrand we call a predictable linear process. We develop basic properties of stochastic integrals against "noncommutative $L^p$-integrators" and establish a useful criterion for a noncommutative stochastic process to be a noncommutative $L^p$-integrator. This criterion applies, in particular, to a rich class of processes we call measured decomposable processes, which includes free Brownian motion and, more generally, the $q$-Brownian motions. We also show that, when specialized appropriately, our framework recovers the classical notion of an $L^p$-integrator up to the fact that the noncommutative theory sees only the modification class of a classical stochastic process. As an application, we use our theory to study noncommutative stochastic differential equations (SDEs) in $L^p$. Under natural local-Lipschitz and boundedness assumptions, we establish the existence and uniqueness of maximal solutions and show that a maximal solution with finite lifetime must blow up in $L^p$ norm. As corollaries, we obtain new existence and uniqueness results for free SDEs and SDEs driven by $q$-Brownian motion. Notably, in the free case, our results apply to coefficients arising through the continuous functional calculus from merely locally Lipschitz scalar functions, as opposed to locally operator-Lipschitz functions.
Comments66 pages, 1 figure
