Elliptic Stochastic PDEs with polynomial perturbations having a correspondence to Euclidean QFT

ψ(x) + λ : ψ(x) := “ − ∂ 2 ∂t2 + (−∆3 + m) ” 1 2 Ẇ (x), (1.1) x ≡ (t, ~x) ∈ R × R, where ∆d is the d-dimensional Laplace operator, W is an isonormal Gaussian process on R , λ ≥ 0 is some given number and : ψ : is the cubic Wick power of ψ. In order to understand an importance and a motivation of the setting of (1,1), we start with the review of (1.2) below for general d ∈ N, which has been considered in [AY1] in a framework of change of variable formula on Nelson’s Euclidean free field: (−∆d + m)ψ(x) + λ : ψ(x) := (−∆d + m) 1 2 Ẇ (x), x ≡ (t, ~x) ∈ R × R, (1.2) where W is an isonormal Gaussian process on R. We have to recall that Nelson’s Euclidean free field is a Gaussian random variable φω taking values in S ′(Rd) defined on a probability space (Ω,F , P ) such that