An SPDE approach to perturbation theory of $Phi^4_2$: asymptoticity and short distance behavior

In this paper we study the perturbation theory of $Phi^4_2$ model on the whole plane via stochastic quantization. We use integration by parts formula (i.e. Dyson-Schwinger equations) to generate the perturbative expansion for the $k$-point correlation functions, and prove bounds on the remainder of the truncated expansion using PDE estimates; this in particular proves that the expansion is asymptotic. Furthermore, we derive short distance behaviors of the $2$-point function and the connected $4$-point function, also via suitable Dyson-Schwinger equations combined with PDE arguments.