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Stochastic Lie group-valued measures and their relations to stochastic curve integrals, gauge fields and Markov cosurfaces. (English) Zbl 0575.60068

Stochastic processes - mathematics and physics, Proc. 1st BiBoS-Symp., Bielefeld/Ger. 1984, Lect. Notes Math. 1158, 1-24 (1986).
[For the entire collection see Zbl 0572.00014.]
We discuss an extension of stochastic analysis to the case where time is multi-dimensional and the state space is a (Lie) group. In particular we study stochastic group-valued measures and generalized semigroups and show how they can be obtained by multiplicative stochastic integration from vector-valued stochastic Lévy-Khinchin fields. We also discuss their connections to group-valued Markov cosurfaces and, in the case of 2-dimensional ”time”, group-valued curve integrals. We analyze furthermore, in the general multi-dimensional case, the relation with curve integrals, connections and gauge fields and mention the application of group-valued Markov cosurfaces to the construction of relativistic fields.

MSC:

60H99 Stochastic analysis
81P20 Stochastic mechanics (including stochastic electrodynamics)
60J99 Markov processes

Citations:

Zbl 0572.00014