Francesco C. De Vecchi, Luca Fresta, Maria Gordina and Massimiliano Gubinelli
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 $\Phi^{4}_{2}$ 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 $\Phi^{4}_{2}$.
Francesco C. De Vecchi, Luca Fresta, Maria Gordina and Massimiliano Gubinelli. Non-commutative $L^{p}$ spaces and Grassmann stochastic analysis. Papers of Massimiliano Gubinelli, 2023, version 1. https://mgubi.org/papers/paper/noncommutative-lp-grassmann/
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