Reflection Positivity and Stochastic Processes

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


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 \(\widehat{\mathscr {E}}\).

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© 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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