Stochastic quantization of $\lambda \phi_2^4$- theory in 2-d Moyal space

There is strong evidence for the conjecture that the $\lambda \phi^4$ QFT- model on 4-dimensional non-commutative Moyal space can be non-perturbatively constructed. As preparation, in this paper we construct the 2-dimensional case with the method of stochastic quantization. We show the local well-posedness and global well-posedness of the stochastic quantization equation, leading to a construction of the Moyal $\lambda \phi^4_2$ measure for any non-negative coupling constant $\lambda$.