Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients

In this paper we develop a new weak convergence and compact embedding method to study the existence and uniqueness of the $L_{\rho}^2({$\mathbb{R}$^{d}};{$\mathbb{R}$^{1}})\otimes L_{\rho}^2({$\mathbb{R}$^{d}};{$\mathbb{R}$^{d}})$ valued solution of backward stochastic differential equations with p-growth coefficients. Then we establish the probabilistic representation of the weak solution of PDEs with p-growth coefficients via corresponding BSDEs.