"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding solutions of these stochastic PDEs, one of the remarkable examples being Hairer and Mourrat-Weber's results on the Phi^4_3 equation.
In this talk we will discuss stochastic quantization of quantum field theory with gauge symmetries, with focus on an Abelian example but also provide prospects of non-Abelian Yang-Mills theories. We address issues regarding Wilson’s lattice regularization, dynamical gauge fixing, renormalization, Ward identities, and construction of dynamical loop and string observables.