  We prove the existence, uniqueness, and strict positivity of ground states of
the possibly massless renormalized Nelson operator under an infrared regularity
condition and for Kato decomposable electrostatic potentials fulfilling a
binding condition. If the infrared regularity condition is violated, then we
show non-existence of ground states of the massless renormalized Nelson
operator with an arbitrary Kato decomposable potential. Furthermore, we prove
the existence, uniqueness, and strict positivity of ground states of the
massless renormalized Nelson operator in a non-Fock representation where the
infrared condition is unnecessary. Exponential and superexponential estimates
on the pointwise spatial decay and the decay with respect to the boson number
for elements of spectral subspaces below localization thresholds are provided.
Moreover, some continuity properties of ground state eigenvectors are
discussed. Byproducts of our analysis are a hypercontractivity bound for the
semigroup and a new remark on Nelson's operator theoretic renormalization
procedure. Finally, we construct path measures associated with ground states of
the renormalized Nelson operator. Their analysis entails improved boson number
decay estimates for ground state eigenvectors, as well as upper and lower
bounds on the Gaussian localization with respect to the field variables in the
ground state. As our results on uniqueness, positivity, and path measures
exploit the ergodicity of the semigroup, we restrict our attention to one
matter particle. All results are non-perturbative.
