Non-commutative L^p spaces and Grassmann stochastic analysis

2023/05/15 by Francesco C. De Vecchi, De Vecchi, Francesco C., Luca Fresta +5
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2305.08497

openalex publication_date 2023/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a theory of non-commutative L^p spaces suitable for non-commutative probability in a non-tracial setting and use it to develop stochastic analysis of Grassmann-valued processes, including martingale inequalities, stochastic integrals with respect to Grassmann Itô processes, Girsanov's formula and a weak formulation of Grassmann SDEs. We apply this new setting to the construction of several unbounded random variables including a Grassmann analog of the Φ⁴₂ Euclidean QFT in a bounded region and weak solution to singular SPDEs in the spirit of the early work of Jona-Lasinio and Mitter on the stochastic quantisation of Φ⁴₂.

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