Continuity of sample paths and weak compactness of measures in Euclidean field theory
Zbigniew Haba · Journal of Mathematical Physics · 1982
Under an assumption on the asymptotic u→∞ behavior of the generating functional ℱ exp[uφ( g)] dμ(φ), an estimate is obtained on ‖χ(x)−χ(x′)‖, where χ(x) = φg(x) ≡φ( g(⋅−x)), showing Hölder continuity of φg(x) for a class of g. It is proved that the family of measures vγ, with dvγ(χ) = dμγ(φg) and ℱ expχ(x) dvγ(χ) bounded in γ, is weakly conditionally compact. In application to the infinite volume cutoff μκ measures in P(φ)d , these results imply the continuity of μκ(C) when κ→∞ for a class of sets C. Such a property allows distinguishing the interacting measures from the free ones.