Noncommutative L^p-integrators and stochastic differential equations in L^p

@misc{jekel2026noncommutative,
  title = {{Noncommutative $L^p$-integrators and stochastic differential equations in $L^p$}},
  author = {David A. Jekel and Todd A. Kemp and Evangelos A. Nikitopoulos},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23948},
  eprint = {2609.23948},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23948},
}

TY  - UNPB
TI  - Noncommutative L^p-integrators and stochastic differential equations in L^p
AU  - David A. Jekel
AU  - Todd A. Kemp
AU  - Evangelos A. Nikitopoulos
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23948
UR  - https://arxiv.org/abs/2609.23948
AB  - 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.
ER  -