Functional Central Limit Theorems and P(ϕ)1-Processes for the Relativistic and Non-Relativistic Nelson Models

Abstract

We construct P (ϕ) 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.

Functional Central Limit Theorem for Additive Functionals Associated to the Generalized Nelson Hamiltonian

“…By the same procedure as in [6] and [7], we show the existence of a stationary reversible Markov process (X t ) t∈R generated by the ground state transformation of H Ψ . This class of process called P (ϕ) 1 -process.…”

“…The main result of this section is to prove a generalized functional central limit theorem for the Nelson model with Bernstein function and to give an explicit expression of the variance. Our main theorem will englobe both classical and relativistic cases discussed in [6] and give rise to a general theorem by introducing a class of functions acting on the particle part.…”