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Reflection Positivity and Stochastic Processes

  • Karl-Hermann NeebEmail author
  • Gestur Ólafsson
Chapter
Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 32)

Abstract

In this chapter we describe some recent generalizations of classical results by Klein and Landau [Kl78, KL75] concerning the interplay between reflection positivity and stochastic processes. Here the main step is the passage from the symmetric semigroup \(({\mathbb R},{\mathbb R}_+,-\mathop {\mathrm{id}}\nolimits _{\mathbb R})\) to more general triples \((G, S,\tau )\). This leads to the concept of a \((G, S,\tau )\)-measure space generalizing Klein’s Osterwalder–Schrader path spaces for \(({\mathbb R},{\mathbb R}_+,-\mathop {\mathrm{id}}\nolimits _{\mathbb R})\). A key result is the correspondence between \((G, S,\tau )\)-measure spaces and the corresponding positive semigroup structures on the Hilbert space \(\widehat{\mathscr {E}}\).

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department MathematikUniversität Erlangen-NürnbergErlangenGermany
  2. 2.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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