  In this paper, we compare dispersions of a scalar field in Euclidean quantum
gravity with stochastic inflation. We use Einstein gravity and a minimally
coupled scalar field with a quadratic potential. We restrict our attention to
small mass and small field cases. In the Euclidean approach, we introduce the
ground state wave function which is approximated by instantons. We used a
numerical technique to find instantons that satisfy classicality. In the
stochastic approach, we introduce the probability distribution of Hubble
patches that can be approximated by locally homogeneous universes down to a
smoothing scale. We assume that the ground state wave function should
correspond to the stationary state of the probability distribution of the
stochastic universe. By comparing the dispersion of both approaches, we
conclude three main results. (1) For a statistical distribution with a certain
value, we can find a corresponding instanton in the Euclidean side, and it
should be a complex-valued instanton. (2) The size of the universe of the
Euclidean approach corresponds to the smoothing scale of the stochastic side;
the universe is homogeneous up to the Euclidean instanton. (3) In addition, as
the mass increases up to a critical value, both approaches break at the same
time. Hence, generation of classical inhomogeneity in the stochastic approach
and the instability of classicality in the Euclidean approach are related.
