Functional central limit theorems and $P(phi)_{1}$-processes for the classical and relativistic Nelson models

TL;DR

This paper constructs $P(\phi)_1$-processes for both relativistic and non-relativistic Nelson models, deriving a central limit theorem for their additive functionals, highlighting differences in path regularity.

Contribution

It introduces a unified construction of $P(\phi)_1$-processes for Nelson models and establishes a CLT for their additive functionals, considering path regularity differences.

Findings

Construction of $P(\phi)_1$-processes for both models

Central limit theorem for additive functionals

Analysis of path regularity differences

Abstract

We construct $P(phi)_1$-processes indexed by the full time-line, separately derived from the functional integral representations of the relativistic and non-relativistic Nelson models in quantum field theory. These two cases differ essentially by sample path regularity. Associated with these processes we define a martingale which, under an appropriate scaling, allows to obtain a central limit theorem for additive functionals of these processes. We discuss a number of examples by choosing specific functionals related to particle-field operators.