# Reflection Positivity and Stochastic Processes

• Karl-Hermann Neeb
• 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 $$\widehat{\mathscr {E}}$$.