Fractional $P(\phi)_1$-processes and Gibbs measures

TL;DR

This paper introduces fractional $P({ ext{phi})}_1$-processes generated by fractional Schrödinger semigroups, proves their existence, and shows their measures are Gibbs measures, analyzing conditions for uniqueness and support.

Contribution

It defines fractional $P({ ext{phi})}_1$-processes, proves their existence, and characterizes their Gibbs measure properties and support conditions.

Findings

Existence of fractional $P({ ext{phi})}_1$-processes established.

Gibbs measure property with respect to the potential proven.

Conditions for uniqueness and support characterized.

Abstract

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.