Non-commutative $$L^p$$ spaces and Grassmann stochastic analysis

We introduce a theory of non-commutative <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi> <mml:mi>p</mml:mi></mml:msup> </mml:math> 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 <i>Itô-Grassmann</i> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>Φ</mml:mi> <mml:mn>2</mml:mn> <mml:mn>4</mml:mn></mml:msubsup> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>Φ</mml:mi> <mml:mn>2</mml:mn> <mml:mn>4</mml:mn></mml:msubsup> </mml:math> .