Fractional $P(φ)_1$-processes and Gibbs measures

Kamil Kaleta, Jozsef Lorinczi

Published 2010-11-11, updated 2011-09-13Version 2

We define and prove existence of fractional $P(\phi)_1$-processes as random processes generated by fractional Schr\"odinger semigroups with Kato-decomposable potentials. Also, we show that the measure of such a process is a Gibbs measure with respect to the same potential. We give conditions of its uniqueness and characterize its support relating this with intrinsic ultracontractivity properties of the semigroup and the fall-off of the ground state. To achieve that we establish and analyze these properties first.